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Principal blocks with 5 irreducible characters

2021-06-11
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
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Unitary $t$-groups

2019-11-25
Eiichi BannaÄ­ , Gabriel Navarro , Noelia Rizo +1 more
Relying on the main results of [GT], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a 
 Relying on the main results of [GT], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary 4-group, which is also a unitary 5-group, but not a unitary $t$-group for any $t \geq 6$.
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Groups with few pâ€Č-character degrees

2020-02-11
Eugenio Giannelli , Noelia Rizo , A. A. Schaeffer Fry
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Groups with few 𝑝’-character degrees in the principal block

2020-04-22
Eugenio Giannelli , Noelia Rizo , Benjamin Sambale +1 more
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≄</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\ge 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper 
 Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≄</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\ge 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group. We prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-solvable of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-length at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if there are at most two distinct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-character degrees in the principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-block of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This generalizes a theorem of Isaacs–Smith as well as a recent result of three of the present authors.
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Nilpotent and perfect groups with the same set of character degrees

2014-03-18
Gabriel Navarro , Noelia Rizo
We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees. We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees.

Characters and generation of Sylow 2-subgroups

2021-02-25
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry +1 more
We show that the character table of a finite group $G$ determines whether a Sylow 2-subgroup of $G$ is generated by 2 elements, in terms of the Galois action on 
 We show that the character table of a finite group $G$ determines whether a Sylow 2-subgroup of $G$ is generated by 2 elements, in terms of the Galois action on characters. Our proof of this result requires the use of the Classification of Finite Simple Groups and provides new evidence for the so-far elusive Alperin–McKay–Navarro conjecture.
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p-Blocks relative to a character of a normal subgroup

2018-08-20
Noelia Rizo
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A Brauer-Wielandt formula (with an application to character tables)

2016-01-20
Gabriel Navarro , Noelia Rizo
If a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts coprimely on 
 If a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts coprimely on a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we give a Brauer-Wielandt formula to count the number of fixed points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue bold upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">C</mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">| \textbf {C}_{G}(P) |</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This serves to determine the number of Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups of certain finite groups from their character tables.

Divisibility of degrees in McKay correspondences

2018-08-31
Noelia Rizo

Galois action on the principal block and cyclic Sylow subgroups

2020-08-18
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
We characterize finite groups G having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block of G, for p " 2, 
 We characterize finite groups G having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block of G, for p " 2, 3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
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Character degrees in separable groups

2021-09-29
Gabriel Navarro , Noelia Rizo , LucĂ­a Sanus
Under some separability conditions, many results on character degrees of finite groups can be unified in a single statement. Under some separability conditions, many results on character degrees of finite groups can be unified in a single statement.

Principal blocks for different primes, I

2022-07-28
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures. We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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Minimal heights and defect groups with two character degrees

2024-02-27
Gunter Malle , Alexander MoretĂł , Noelia Rizo
Conjecture A of [3] predicts the equality between the smallest positive height of the irreducible characters in a p-block of a finite group and the smallest positive height of the 
 Conjecture A of [3] predicts the equality between the smallest positive height of the irreducible characters in a p-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
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p-Blocks Relative to a Character of a Normal Subgroup

2018-02-22
Noelia Rizo
Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we 
 Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this partition a theta-block, and to each theta-block B_theta we naturally associate a conjugacy class of p-subgroups of G/N, which we call the theta-defect groups of B_theta. If N is trivial, then the theta-blocks are the Brauer p-blocks. Using theta-blocks, we can unify the Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every theta-block B_theta has size less than or equal the size of any of its theta-defect groups, hence bringing normal subgroups to the k(B)-conjecture.

Unitary t-groups

2018-10-05
Eiichi BannaÄ­ , Gabriel Navarro , Noelia Rizo +1 more
Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a 
 Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a unitary $t$-group for any $t \geq 6$.

Groups with few $p'$-character degrees

2019-04-13
Eugenio Giannelli , Noelia Rizo , Mandi Schaeffer Fry
We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a 
 We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.

Kernels of $$p'$$-Degree Irreducible Characters

2022-05-03
Alexander MoretĂł , Noelia Rizo
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Brauer character degrees and Sylow normalizers

2022-03-19
Lorenzo Bonazzi , Gabriel Navarro , Noelia Rizo +1 more
Abstract If p and q are primes, and G is a p -solvable finite group, it is possible to detect that a q -Sylow normalizer is contained in a p 
 Abstract If p and q are primes, and G is a p -solvable finite group, it is possible to detect that a q -Sylow normalizer is contained in a p -Sylow normalizer using the character table of G . This is characterized in terms of the degrees of p -Brauer characters. Some consequences, which include yet another generalization of the Itî–Michler theorem, are also obtained.
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Principal blocks with 5 irreducible characters

2020-10-29
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is 
 We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

The blocks with four irreducible characters

2022-11-05
Jorge MartĂ­nez , Noelia Rizo , LucĂ­a Sanus
Abstract Suppose that is a Brauer ‐block of a finite group with defect group . If exactly contains four ordinary irreducible characters, then we show that has order four or 
 Abstract Suppose that is a Brauer ‐block of a finite group with defect group . If exactly contains four ordinary irreducible characters, then we show that has order four or five, assuming the Alperin–McKay conjecture holds for .
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Unitary t-groups

2018-01-01
Eiichi BannaÄ­ , Gabriel Navarro , Noelia Rizo +1 more
Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a 
 Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a unitary $t$-group for any $t \geq 6$.

Certain irreducible characters over a normal subgroup

2016-09-01
Gabriel Navarro , Noelia Rizo
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the 
 We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.

Certain irreducible characters over a normal subgroup

2016-11-06
Gabriel Navarro , Noelia Rizo
Abstract We extend the Howlett–Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all 
 Abstract We extend the Howlett–Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.
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A real version of Thompson’s Theorem on Degrees

2020-09-23
Noelia Rizo , LucĂ­a Sanus

p-Blocks Relative to a Character of a Normal Subgroup II

2020-10-13
Noelia Rizo

Principal blocks for different primes, I

2022-01-01
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures. We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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Principal blocks for different primes, II

2022-01-01
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support 
 If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.
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The blocks with four irreducible characters

2021-01-01
J. Miquel MartĂ­nez , Noelia Rizo , LucĂ­a Sanus
Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the 
 Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.

Groups with few $p'$-character degrees

2019-01-01
Eugenio Giannelli , Noelia Rizo , Mandi Schaeffer Fry
We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a 
 We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.
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Kernels of p'-degree irreducible characters

2021-01-01
Alexander MoretĂł , Noelia Rizo
We prove a p'-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees. We prove a p'-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees.
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Principal blocks with 5 irreducible characters

2020-01-01
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is 
 We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

Groups with few $p'$-character degrees in the principal block

2020-01-01
Eugenio Giannelli , Noelia Rizo , Benjamin Sambale +1 more
Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at 
 Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a theorem of Isaacs-Smith, as well as a recent result of three of the present authors.
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Certain irreducible characters over a normal subgroup

2016-01-01
Gabriel Navarro , Noelia Rizo
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the 
 We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.

p-Blocks Relative to a Character of a Normal Subgroup

2018-01-01
Noelia Rizo
Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we 
 Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this partition a theta-block, and to each theta-block B_theta we naturally associate a conjugacy class of p-subgroups of G/N, which we call the theta-defect groups of B_theta. If N is trivial, then the theta-blocks are the Brauer p-blocks. Using theta-blocks, we can unify the Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every theta-block B_theta has size less than or equal the size of any of its theta-defect groups, hence bringing normal subgroups to the k(B)-conjecture.
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Principal Blocks for Different Primes, II

2022-12-07
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
Abstract If G is a finite group, we have proposed three new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper,we give strong 
 Abstract If G is a finite group, we have proposed three new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper,we give strong support to the validity of Conjectures B and C.
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Minimal heights ands defect groups with two character degrees

2023-01-01
Gunter Malle , Alexander MoretĂł , Noelia Rizo
Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the 
 Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
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The blocks with five irreducible characters

2023-01-01
J. Miquel MartĂ­nez , Noelia Rizo , LucĂ­a Sanus
Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters 
 Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters in $B$ is $5$ then $D\cong C_5, C_7, D_8$ or $Q_8$, assuming that the Alperin--McKay conjecture holds for $B$.
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Brauer's problem 21 for principal blocks

2023-01-01
Alexander MoretĂł , Noelia Rizo , A. A. Schaeffer Fry
Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group 
 Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with 3 irreducible characters is necessarily the cyclic group of order 3. In most cases we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer's height zero conjecture.

The Alperin Weight Conjecture and the Glauberman correspondence via character triples

2023-01-01
Jorge MartĂ­nez , Noelia Rizo , Damiano Rossi
Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple 
 Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that assuming the so-called Inductive (Blockwise) Alperin Weight Condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin Weight Conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro's conjecture for each finite group.

A Brauer--Galois height zero conjecture

2024-02-13
Gunter Malle , Alexander MoretĂł , Noelia Rizo +1 more
Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any 
 Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups.
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Decomposition numbers in the principal block and Sylow normalisers

2024-05-14
Gunter Malle , Noelia Rizo
If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block 
 If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block of G and the p-local structure of G.
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Characters and Sylow $3$-subgroup commutators

2024-05-26
Eugenio Giannelli , Noelia Rizo , A. A. Schaeffer Fry +1 more
We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 
 We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of G, settling a conjecture put forward by Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result by Laradji on the number of character of height zero in a block that lie above a given character of some normal subgroup holds, without any hypothesis on the group for blocks of maximal defect.
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Fixed point ratios, Sylow numbers and coverings of $p$-elements in finite groups

2024-07-29
Robert M. Guralnick , Attila MarĂłti , Juan MartĂ­nez Madrid +2 more
Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime 
 Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.
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Characters and Sylow subgroup abelianization

2025-01-01
Eugenio Giannelli , Noelia Rizo , A. A. Schaeffer Fry +1 more

Brauer’s problem 21 for principal blocks

2025-01-23
Alexander MoretĂł , Noelia Rizo , A. A. Schaeffer Fry
Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are finitely 
 Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are finitely many isomorphism classes of groups that occur as the defect group of a block with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible characters is necessarily the cyclic group of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In most cases, we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer’s height zero conjecture.

A Brauer–Galois Height Zero Conjecture

2025-03-01
Gunter Malle , Alexander MoretĂł , Noelia Rizo +1 more
Abstract Recently, Malle and Navarro provided a Galois-theoretic enhancement of Brauer’s height zero conjecture for principal $p$-blocks, limited to the case $p=2$, utilizing a specific Galois automorphism of order 2. 
 Abstract Recently, Malle and Navarro provided a Galois-theoretic enhancement of Brauer’s height zero conjecture for principal $p$-blocks, limited to the case $p=2$, utilizing a specific Galois automorphism of order 2. While they left open the question of whether a similar result could hold for odd primes, in this paper, we significantly advance their work by formulating a broader Galois version of the conjecture for any prime $p$, using an elementary abelian $p$-subgroup of the absolute Galois group. We not only strengthen their result for $p=2$, but also prove the conjecture for arbitrary primes $p$, except when $G$ contains certain small-rank Lie-type groups as composition factors. Moreover, we establish the conjecture for almost simple groups and for $p$-solvable groups.

Fixed‐point ratios, Sylow numbers, and coverings of p$p$‐elements in finite groups

2025-05-01
Robert M. Guralnick , Attila Maróti , Juan José Prieto Martínez +2 more
Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime 
 Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime and a finite group , we use fixed‐point ratios to study the number of Sylow ‐subgroups of and the minimal size of a covering by proper subgroups of the set of ‐elements of .

Decomposition numbers in the principal block and Sylow normalisers

2025-05-26
Gunter Malle , Noelia Rizo

Decomposition numbers in the principal block and Sylow normalisers

2025-05-26
Gunter Malle , Noelia Rizo

Fixed‐point ratios, Sylow numbers, and coverings of p$p$‐elements in finite groups

2025-05-01
Robert M. Guralnick , Attila Maróti , Juan José Prieto Martínez +2 more
Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime 
 Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime and a finite group , we use fixed‐point ratios to study the number of Sylow ‐subgroups of and the minimal size of a covering by proper subgroups of the set of ‐elements of .

A Brauer–Galois Height Zero Conjecture

2025-03-01
Gunter Malle , Alexander MoretĂł , Noelia Rizo +1 more
Abstract Recently, Malle and Navarro provided a Galois-theoretic enhancement of Brauer’s height zero conjecture for principal $p$-blocks, limited to the case $p=2$, utilizing a specific Galois automorphism of order 2. 
 Abstract Recently, Malle and Navarro provided a Galois-theoretic enhancement of Brauer’s height zero conjecture for principal $p$-blocks, limited to the case $p=2$, utilizing a specific Galois automorphism of order 2. While they left open the question of whether a similar result could hold for odd primes, in this paper, we significantly advance their work by formulating a broader Galois version of the conjecture for any prime $p$, using an elementary abelian $p$-subgroup of the absolute Galois group. We not only strengthen their result for $p=2$, but also prove the conjecture for arbitrary primes $p$, except when $G$ contains certain small-rank Lie-type groups as composition factors. Moreover, we establish the conjecture for almost simple groups and for $p$-solvable groups.

Brauer’s problem 21 for principal blocks

2025-01-23
Alexander MoretĂł , Noelia Rizo , A. A. Schaeffer Fry
Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are finitely 
 Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are finitely many isomorphism classes of groups that occur as the defect group of a block with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible characters is necessarily the cyclic group of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In most cases, we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer’s height zero conjecture.

Characters and Sylow subgroup abelianization

2025-01-01
Eugenio Giannelli , Noelia Rizo , A. A. Schaeffer Fry +1 more

Fixed point ratios, Sylow numbers and coverings of $p$-elements in finite groups

2024-07-29
Robert M. Guralnick , Attila MarĂłti , Juan MartĂ­nez Madrid +2 more
Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime 
 Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.
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Characters and Sylow $3$-subgroup commutators

2024-05-26
Eugenio Giannelli , Noelia Rizo , A. A. Schaeffer Fry +1 more
We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 
 We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of G, settling a conjecture put forward by Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result by Laradji on the number of character of height zero in a block that lie above a given character of some normal subgroup holds, without any hypothesis on the group for blocks of maximal defect.
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Decomposition numbers in the principal block and Sylow normalisers

2024-05-14
Gunter Malle , Noelia Rizo
If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block 
 If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block of G and the p-local structure of G.
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Minimal heights and defect groups with two character degrees

2024-02-27
Gunter Malle , Alexander MoretĂł , Noelia Rizo
Conjecture A of [3] predicts the equality between the smallest positive height of the irreducible characters in a p-block of a finite group and the smallest positive height of the 
 Conjecture A of [3] predicts the equality between the smallest positive height of the irreducible characters in a p-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
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A Brauer--Galois height zero conjecture

2024-02-13
Gunter Malle , Alexander MoretĂł , Noelia Rizo +1 more
Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any 
 Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups.
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Minimal heights ands defect groups with two character degrees

2023-01-01
Gunter Malle , Alexander MoretĂł , Noelia Rizo
Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the 
 Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
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The blocks with five irreducible characters

2023-01-01
J. Miquel MartĂ­nez , Noelia Rizo , LucĂ­a Sanus
Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters 
 Let $G$ be a finite group, $p$ a prime and $B$ a Brauer $p$-block of $G$ with defect group $D$. We prove that if the number of irreducible ordinary characters in $B$ is $5$ then $D\cong C_5, C_7, D_8$ or $Q_8$, assuming that the Alperin--McKay conjecture holds for $B$.
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Brauer's problem 21 for principal blocks

2023-01-01
Alexander MoretĂł , Noelia Rizo , A. A. Schaeffer Fry
Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group 
 Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with 3 irreducible characters is necessarily the cyclic group of order 3. In most cases we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer's height zero conjecture.

The Alperin Weight Conjecture and the Glauberman correspondence via character triples

2023-01-01
Jorge MartĂ­nez , Noelia Rizo , Damiano Rossi
Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple 
 Recently, G. Navarro introduced a new conjecture that unifies the Alperin Weight Conjecture and the Glauberman correspondence into a single statement. In this paper, we reduce this problem to simple groups and prove it for several classes of groups and blocks. Our reduction can be divided into two steps. First, we show that assuming the so-called Inductive (Blockwise) Alperin Weight Condition for finite simple groups, we obtain an analogous statement for arbitrary finite groups, that is, an automorphism-equivariant version of the Alperin Weight Conjecture inducing isomorphisms of modular character triples. Then, we show that the latter implies Navarro's conjecture for each finite group.

Principal Blocks for Different Primes, II

2022-12-07
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
Abstract If G is a finite group, we have proposed three new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper,we give strong 
 Abstract If G is a finite group, we have proposed three new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper,we give strong support to the validity of Conjectures B and C.
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The blocks with four irreducible characters

2022-11-05
Jorge MartĂ­nez , Noelia Rizo , LucĂ­a Sanus
Abstract Suppose that is a Brauer ‐block of a finite group with defect group . If exactly contains four ordinary irreducible characters, then we show that has order four or 
 Abstract Suppose that is a Brauer ‐block of a finite group with defect group . If exactly contains four ordinary irreducible characters, then we show that has order four or five, assuming the Alperin–McKay conjecture holds for .
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Principal blocks for different primes, I

2022-07-28
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures. We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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Kernels of $$p'$$-Degree Irreducible Characters

2022-05-03
Alexander MoretĂł , Noelia Rizo
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Brauer character degrees and Sylow normalizers

2022-03-19
Lorenzo Bonazzi , Gabriel Navarro , Noelia Rizo +1 more
Abstract If p and q are primes, and G is a p -solvable finite group, it is possible to detect that a q -Sylow normalizer is contained in a p 
 Abstract If p and q are primes, and G is a p -solvable finite group, it is possible to detect that a q -Sylow normalizer is contained in a p -Sylow normalizer using the character table of G . This is characterized in terms of the degrees of p -Brauer characters. Some consequences, which include yet another generalization of the Itî–Michler theorem, are also obtained.
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Principal blocks for different primes, I

2022-01-01
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures. We propose new conjectures about the relationship between the principal blocks of finite groups for different primes and establish evidence for these conjectures.
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Principal blocks for different primes, II

2022-01-01
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry
If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support 
 If G is a finite group, we have proposed new conjectures on the interaction between different primes and their corresponding Brauer principal blocks. In this paper, we give strong support to the validity of these conjectures.
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Character degrees in separable groups

2021-09-29
Gabriel Navarro , Noelia Rizo , LucĂ­a Sanus
Under some separability conditions, many results on character degrees of finite groups can be unified in a single statement. Under some separability conditions, many results on character degrees of finite groups can be unified in a single statement.

Principal blocks with 5 irreducible characters

2021-06-11
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
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Characters and generation of Sylow 2-subgroups

2021-02-25
Gabriel Navarro , Noelia Rizo , A. A. Schaeffer Fry +1 more
We show that the character table of a finite group $G$ determines whether a Sylow 2-subgroup of $G$ is generated by 2 elements, in terms of the Galois action on 
 We show that the character table of a finite group $G$ determines whether a Sylow 2-subgroup of $G$ is generated by 2 elements, in terms of the Galois action on characters. Our proof of this result requires the use of the Classification of Finite Simple Groups and provides new evidence for the so-far elusive Alperin–McKay–Navarro conjecture.
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The blocks with four irreducible characters

2021-01-01
J. Miquel MartĂ­nez , Noelia Rizo , LucĂ­a Sanus
Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the 
 Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.

Kernels of p'-degree irreducible characters

2021-01-01
Alexander MoretĂł , Noelia Rizo
We prove a p'-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees. We prove a p'-version of a classical theorem of Broline and Garrison. As a consequence, we obtain results on p-parts of character codegrees.
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Principal blocks with 5 irreducible characters

2020-10-29
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is 
 We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

p-Blocks Relative to a Character of a Normal Subgroup II

2020-10-13
Noelia Rizo

A real version of Thompson’s Theorem on Degrees

2020-09-23
Noelia Rizo , LucĂ­a Sanus

Galois action on the principal block and cyclic Sylow subgroups

2020-08-18
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
We characterize finite groups G having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block of G, for p " 2, 
 We characterize finite groups G having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block of G, for p " 2, 3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
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Groups with few 𝑝’-character degrees in the principal block

2020-04-22
Eugenio Giannelli , Noelia Rizo , Benjamin Sambale +1 more
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≄</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\ge 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper 
 Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≄</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\ge 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group. We prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-solvable of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-length at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if there are at most two distinct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-character degrees in the principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-block of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This generalizes a theorem of Isaacs–Smith as well as a recent result of three of the present authors.
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Groups with few pâ€Č-character degrees

2020-02-11
Eugenio Giannelli , Noelia Rizo , A. A. Schaeffer Fry
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Principal blocks with 5 irreducible characters

2020-01-01
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is 
 We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

Groups with few $p'$-character degrees in the principal block

2020-01-01
Eugenio Giannelli , Noelia Rizo , Benjamin Sambale +1 more
Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at 
 Let p be a prime larger than 3 and let G be a finite group. We prove that G is p-solvable of p-length at most 2 if there are at most two distinct character degrees relatively prime to p in the principal p-block of G. This generalizes a theorem of Isaacs-Smith, as well as a recent result of three of the present authors.
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Unitary $t$-groups

2019-11-25
Eiichi BannaÄ­ , Gabriel Navarro , Noelia Rizo +1 more
Relying on the main results of [GT], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a 
 Relying on the main results of [GT], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary 4-group, which is also a unitary 5-group, but not a unitary $t$-group for any $t \geq 6$.
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Groups with few $p'$-character degrees

2019-04-13
Eugenio Giannelli , Noelia Rizo , Mandi Schaeffer Fry
We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a 
 We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.

Groups with few $p'$-character degrees

2019-01-01
Eugenio Giannelli , Noelia Rizo , Mandi Schaeffer Fry
We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a 
 We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.
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Unitary t-groups

2018-10-05
Eiichi BannaÄ­ , Gabriel Navarro , Noelia Rizo +1 more
Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a 
 Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a unitary $t$-group for any $t \geq 6$.

Divisibility of degrees in McKay correspondences

2018-08-31
Noelia Rizo

p-Blocks relative to a character of a normal subgroup

2018-08-20
Noelia Rizo
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p-Blocks Relative to a Character of a Normal Subgroup

2018-02-22
Noelia Rizo
Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we 
 Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this partition a theta-block, and to each theta-block B_theta we naturally associate a conjugacy class of p-subgroups of G/N, which we call the theta-defect groups of B_theta. If N is trivial, then the theta-blocks are the Brauer p-blocks. Using theta-blocks, we can unify the Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every theta-block B_theta has size less than or equal the size of any of its theta-defect groups, hence bringing normal subgroups to the k(B)-conjecture.

Unitary t-groups

2018-01-01
Eiichi BannaÄ­ , Gabriel Navarro , Noelia Rizo +1 more
Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a 
 Relying on the main results of [Guralnick-Tiep], we classify all unitary $t$-groups for $t \geq 2$ in any dimension $d \geq 2$. We also show that there is essentially a unique unitary $4$-group, which is also a unitary $5$-group, but not a unitary $t$-group for any $t \geq 6$.

p-Blocks Relative to a Character of a Normal Subgroup

2018-01-01
Noelia Rizo
Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we 
 Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this partition a theta-block, and to each theta-block B_theta we naturally associate a conjugacy class of p-subgroups of G/N, which we call the theta-defect groups of B_theta. If N is trivial, then the theta-blocks are the Brauer p-blocks. Using theta-blocks, we can unify the Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every theta-block B_theta has size less than or equal the size of any of its theta-defect groups, hence bringing normal subgroups to the k(B)-conjecture.
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Certain irreducible characters over a normal subgroup

2016-11-06
Gabriel Navarro , Noelia Rizo
Abstract We extend the Howlett–Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all 
 Abstract We extend the Howlett–Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.
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Certain irreducible characters over a normal subgroup

2016-09-01
Gabriel Navarro , Noelia Rizo
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the 
 We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.

A Brauer-Wielandt formula (with an application to character tables)

2016-01-20
Gabriel Navarro , Noelia Rizo
If a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts coprimely on 
 If a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acts coprimely on a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we give a Brauer-Wielandt formula to count the number of fixed points <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue bold upper C Subscript upper G Baseline left-parenthesis upper P right-parenthesis EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">C</mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">| \textbf {C}_{G}(P) |</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This serves to determine the number of Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups of certain finite groups from their character tables.

Certain irreducible characters over a normal subgroup

2016-01-01
Gabriel Navarro , Noelia Rizo
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the 
 We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.

Nilpotent and perfect groups with the same set of character degrees

2014-03-18
Gabriel Navarro , Noelia Rizo
We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees. We find a pair of finite groups, one nilpotent and the other perfect, with the same set of character degrees.
Coauthor Papers Together
A. A. Schaeffer Fry 18
Gabriel Navarro 15
Alexander MoretĂł 10
Eugenio Giannelli 7
Carolina Vallejo 7
Gunter Malle 6
LucĂ­a Sanus 6
Eiichi BannaÄ­ 3
Pham Huu Tiep 3
Attila MarĂłti 2
Benjamin Sambale 2
Mandi Schaeffer Fry 2
Robert M. Guralnick 2
Jorge MartĂ­nez 2
J. Miquel MartĂ­nez 2
Lorenzo Bonazzi 1
Juan José Prieto Martínez 1
Juan MartĂ­nez Madrid 1
Damiano Rossi 1

Character theory of finite groups

1999-07-01
I. M. Isaacs
Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur 
 Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.
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Extensions of unipotent characters and the inductive McKay condition

2008-07-27
Gunter Malle
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Height 0 characters of finite groups of Lie type

2007-12-05
Gunter Malle
We give a classification of irreducible characters of finite groups of Lie type of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree, 
 We give a classification of irreducible characters of finite groups of Lie type of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any prime different from the defining characteristic, in terms of local data. More precisely, we give a classification in terms of data related to the normalizer of a suitable Levi subgroup, which in many cases coincides with the normalizer of a Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroup. The McKay conjecture asserts that there exists a bijection between characters of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree of a group and of the normalizer of a Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroup. We hope that our result will constitute a major step towards a proof of this conjecture for groups of Lie type, and, in conjunction with a recent reduction result of Isaacs, Malle and Navarro, for arbitrary finite groups.

Remarks on isomorphic blocks

1977-03-01
Everett C. Dade

Brauer characters with cyclotomic field of values

2007-09-05
Gabriel Navarro , Pham Huu Tiep , Alexandre Turull

Blocks of Finite Groups and Their Invariants

2014-01-01
Benjamin Sambale

Characters of relative p'-degree over normal subgroups

2013-08-02
Gabriel Navarro , Pham Huu Tiep
Let Z be a normal subgroup of a finite group G, let λ ∈ Irr(Z) be an irreducible complex character of Z, and let p be a prime number.If p 
 Let Z be a normal subgroup of a finite group G, let λ ∈ Irr(Z) be an irreducible complex character of Z, and let p be a prime number.If p does not divide the integers χ(1)/λ(1) for all χ ∈ Irr(G) lying over λ, then we prove that the Sylow p-subgroups of G/Z are abelian.This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture.
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Finite Group Theory

2008-08-06
I. M. Isaacs

Regular and Semisimple Blocks of Finite Reductive Groups

1990-02-01
Gerhard Hiß

The blocks of finite general linear and unitary groups

1982-02-01
Paul Fong , Bhama Srinivasan

Sur les l-blocs unipotents des groupes réductifs finis quand l est mauvais

2000-08-01
Michel Enguehard

The sylow 2-subgroups of the finite classical groups

1964-07-01
Roger Carter , Paul Fong

Principal blocks with 5 irreducible characters

2021-06-11
Noelia Rizo , A. A. Schaeffer Fry , Carolina Vallejo
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Quasi-isolated blocks and Brauer's height zero conjecture

2013-04-04
Radha Kessar , Gunter Malle
This paper has two main results.Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for 
 This paper has two main results.Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes.This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups.Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation.We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups.
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Characters and Sylow 2-subgroups of maximal class revisited

2018-02-09
Gabriel Navarro , Benjamin Sambale , Pham Huu Tiep
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On the p-blocks of symmetric and alternating groups and their covering groups

1990-01-01
JĂžrn B. Olsson

CHEVIE ? A system for computing and processing generic character tables

1996-05-01
Meinolf Geck , Gerhard Hiß , Frank Lïżœbeck +2 more

On the number of irreducible characters in a 2-block

1981-02-01
Peter Landrock

On the Degrees of the Irreducible Representations of Symmetric Groups

1971-07-01
I. G. Macdonald

The blocks of finite classical groups.

1989-05-01
Paul Fong , Bhama Srinivasan

Character correspondences in finite general linear, unitary and symmetric groups

1983-06-01
Gerhard O. Michler , Jïżœrn B. Olsson

Variations on the ItĂŽ-Michler theorem on character degrees

2016-08-01
Gabriel Navarro
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FINITE GROUPS OF LIE TYPE Conjugacy classes and complex characters

1987-05-01
J. A. Green
By Roger W. Carter: pp. 544. ÂŁ42.50. (John Wiley & Sons Ltd, 1985) By Roger W. Carter: pp. 544. ÂŁ42.50. (John Wiley & Sons Ltd, 1985)

Above the Glauberman correspondence

2007-12-01
Alexandre Turull

Die unipotenten charaktere von <sup>2</sup>F<sub>4</sub>(q<sup>2</sup>)

1990-01-01
Gunter Malle
.Alle endlichen Gruppen vom Lie-Typ besitzen eine ausgezeichnete Menge von irreduziblen komplexen Charakteren, die von Lusztig eingefĂŒhrten unipotenten Charaktere. Diese werden fĂŒlr einen vorgegebenen Typ unabhĂ€ngig vom zugrunde liegenden Körper 
 .Alle endlichen Gruppen vom Lie-Typ besitzen eine ausgezeichnete Menge von irreduziblen komplexen Charakteren, die von Lusztig eingefĂŒhrten unipotenten Charaktere. Diese werden fĂŒlr einen vorgegebenen Typ unabhĂ€ngig vom zugrunde liegenden Körper klassifiziert. Aufgrund der ebenfalls von Lusztig bewiesenen Jordan-Zerlegung der irreduziblen komplexen Charaktere laiit sich die Charaktertafel einer jeden endlichen Gruppe G vom Lie-Typ im wesentlichen zusammenbauen, wenn man die unipotenten Charaktere der halbeinfachen Anteile der Zentralisatoren aller halbeinfachen Elemente in G kennt. Das Problem der Berechnung der Charaktertafel ist somit zurĂŒckgefĂŒhrt auf die Bestimmung aller Werte der unipotenten Charaktere der einfachen Gruppen vom Lie-Typ.

Characters of 𝜋’-degree

2019-04-03
Eugenio Giannelli , A. A. Schaeffer Fry , Carolina Vallejo
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> 
 Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a set of primes. Write <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I r r Subscript pi prime Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Irr</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>⁥</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Irr}_{\pi ’}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the set of irreducible characters of degree not divisible by any prime in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π</mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains at most two prime numbers and the only element in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I r r Subscript pi prime Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Irr</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>⁥</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Irr}_{\pi ’}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the principal character, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals 1"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">G=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Finite groups with two conjugacy classes of<i>p</i>-elements and related questions for<i>p</i>-blocks

2014-01-12
Burkhard KĂŒlshammer , Gabriel Navarro , Benjamin Sambale +1 more
We prove that a finite group in which any two nontrivial p-elements are conjugate have Sylow p-subgroups which are either elementary abelian or extra-special of order p 3 and exponent 
 We prove that a finite group in which any two nontrivial p-elements are conjugate have Sylow p-subgroups which are either elementary abelian or extra-special of order p 3 and exponent p.

𝑝-rational characters and self-normalizing Sylow 𝑝-subgroups

2007-04-19
Gabriel Navarro , Pham Huu Tiep , Alexandre Turull
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 
 Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite group, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a prime, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the irreducible characters of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">N</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {N}_G(P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which preserves field of values of correspondent characters (over the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no non-trivial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p prime"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>â€Č</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">p’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational irreducible characters if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper N Subscript upper G Baseline left-parenthesis upper P right-parenthesis equals upper P"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">N</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {N}_G(P)=P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Symmetric local algebras and small blocks of finite groups

1984-05-01
Burkhard KĂŒlshammer

Inductive McKay condition in defining characteristic

2011-11-15
Britta SpÀth
We reformulate the inductive McKay condition, from Isaacs-Malle-Navarro, and apply the new criterion to simple groups of Lie type, when the prime is the defining characteristic p. Thereby we make 
 We reformulate the inductive McKay condition, from Isaacs-Malle-Navarro, and apply the new criterion to simple groups of Lie type, when the prime is the defining characteristic p. Thereby we make use of a recent result of Maslowski. This proves that these simple group satisfy the inductive McKay condition for p. In the non simply-laced types and non-classical types this reproves earlier results by Brunat and Brunat-Himstedt.
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Character table and blocks of finite simple triality groups Âłđ·â‚„(𝑞)

1987-01-01
D. I. Deriziotis , Gerhard O. Michler
Based on recent work of Spaltenstein [<bold>14</bold>] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple 
 Based on recent work of Spaltenstein [<bold>14</bold>] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="cubed upper D 4 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}^3{D_4}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given. As an application we obtain a classification of the irreducible characters of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="cubed upper D 4 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}^3{D_4}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-blocks for all primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This enables us to verify Brauer’s height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="cubed upper D 4 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}^3{D_4}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It also follows that for every prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are blocks of defect zero in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="cubed upper D 4 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{}^3{D_4}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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A lower bound for the number of irreducible characters in a block

1982-02-01
JĂžrgen Brandt

Action of automorphisms on irreducible characters of symplectic groups

2018-03-19
Jay Taylor
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Brauer's height zero conjecture for the 2-blocks of maximal defect

2011-09-26
Gabriel Navarro , Pham Huu Tiep
We prove Brauer's height zero conjecture on blocks of finite groups for every 2-block of maximal defect. We prove Brauer's height zero conjecture on blocks of finite groups for every 2-block of maximal defect.

Quasi-Isolated Elements in Reductive Groups

2005-06-01
Cédric Bonnafé
Abstract A semisimple element s of a connected reductive group G is called quasi-isolated (respectively isolated) if C G ( s ) (respectively ( s )) is not contained in 
 Abstract A semisimple element s of a connected reductive group G is called quasi-isolated (respectively isolated) if C G ( s ) (respectively ( s )) is not contained in a Levi subgroup of a proper parabolic subgroup of G . We study properties of quasi-isolated semisimple elements and give a classification in terms of the affine Dynkin diagram of G . Tables are provided for adjoint simple groups. Key Words: Affine Dynkin diagramIsolated elementReductive groupsSemisimple element1991 Mathematics Subject Classification: Primary 20G99Secondary 20E45 ACKNOWLEDGMENTS We wish to thank F. Digne, J. Michel and M. Enguehard for their valuable comments on this work. Notes #Communicated by J. Alev.
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The McKay conjecture for exceptional groups and odd primes

2008-04-14
Britta SpÀth

A note on groups of p-length 1∗

1976-02-01
I. M. Isaacs , Stephen D. Smith

On the number of characters in blocks of finite general linear, unitary and symmetric groups

1984-03-01
Jïżœrn B. Olsson

Characters of Solvable and Symplectic Groups

1973-01-01
I. M. Isaacs

A Frobenius theorem for blocks

1980-02-01
Michel Broué , Lluís Puig

The McKay conjecture and Galois automorphisms

2004-11-01
Gabriel Navarro
The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group G can be computed locally.The 
 The main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group G can be computed locally.The simplest of these conjectures is the "McKay conjecture" which asserts that the number of irreducible complex characters of G of degree not divisible by p is the same if computed in a p-Sylow normalizer of G.In this paper, we propose a much stronger version of this conjecture which deals with Galois automorphisms.In fact, the same idea can be applied to the celebrated Alperin and Dade conjectures.
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Number of Sylow subgroups in $p$-solvable groups

2003-03-11
Gabriel Navarro
If $G$ is a finite group and $p$ is a prime number, let $\nu _p(G)$ be the number of Sylow $p$-subgroups of $G$. If $H$ is a subgroup of a 
 If $G$ is a finite group and $p$ is a prime number, let $\nu _p(G)$ be the number of Sylow $p$-subgroups of $G$. If $H$ is a subgroup of a $p$-solvable group $G$, we prove that $\nu _p(H)$ divides $\nu _p(G)$.
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The Affine Permutation Groups of Rank Three

1987-05-01
Martin W. Liebeck

Normal p-complements and irreducible characters

1970-02-01
John G. Thompson

Characters of Π-separable groups

1984-01-01
I. M. Isaacs

Regular elements of finite reflection groups

1974-06-01
T. A. Springer

Twisted group algebras and their representations

1964-05-01
Stephen Conlon
Let be a finite group, a field. A twisted group algebra A ( ) on over is an associative algebra whose elements are the formal linear combinations and in which 
 Let be a finite group, a field. A twisted group algebra A ( ) on over is an associative algebra whose elements are the formal linear combinations and in which the product (A)(B) is a non-zero multiple of ( AB ), where AB is the group product of A, B ∈ : . One gets the ordinary group algebra ( ) by taking each f A, B ≠ 1.
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Characters of Relatively Projective Modules II

1987-08-01
Burkhard KĂŒlshammer , Geoffrey R. Robinson

3-blocks and 3-modular characters of G2(q)

1990-06-01
Gerhard Hiß , Josephine Shamash

On the inductive Alperin–McKay and Alperin weight conjecture for groups with abelian Sylow subgroups

2013-09-28
Gunter Malle