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This chapter contains sections titled: Polynomials of Prime Degree Imprimitive Polynomials of Prime-Squared Degree Primitive Permutation Groups Primitive Polynomials of Prime-Squared Degree
This chapter contains sections titled: Polynomials of Prime Degree Imprimitive Polynomials of Prime-Squared Degree Primitive Permutation Groups Primitive Polynomials of Prime-Squared Degree
Let $G$ be a non-abelian group and let $Z(G)$ be the center of $G$. Associate with $G$ there is agraph $Gamma_G$ as follows: Take $Gsetminus Z(G)$ as vertices of$Gamma_G$ and …
Let $G$ be a non-abelian group and let $Z(G)$ be the center of $G$. Associate with $G$ there is agraph $Gamma_G$ as follows: Take $Gsetminus Z(G)$ as vertices of$Gamma_G$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $Gamma_G$ is called the non-commuting graph of $G$. In recent years many interesting works have been done in non-commutative graph of groups. Computing the clique number, chromatic number, Szeged index and Wiener index play important role in graph theory. In particular, the clique number of non-commuting graph of some the general linear groups has been determined. nt Recently, Wiener and Szeged indiceshave been computed for $Gamma_{PSL(2,q)}$, where $qequiv 0 (mod~~4)$. In this paper we will compute the Szeged index for$Gamma_{PSL(2,q)}$, where $qnotequiv 0 (mod ~~ 4)$.
Let [Formula: see text] be a finite group, [Formula: see text] be a partition of the set of all primes [Formula: see text] and [Formula: see text]. A set [Formula: …
Let [Formula: see text] be a finite group, [Formula: see text] be a partition of the set of all primes [Formula: see text] and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] possesses a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]. Let [Formula: see text] be a subgroup of [Formula: see text]. [Formula: see text] is: [Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for some modular subgroup [Formula: see text] and [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text]; weakly[Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if there are an [Formula: see text]-[Formula: see text]-permutable subgroup [Formula: see text] and a [Formula: see text]-subnormal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we investigate the influence of weakly [Formula: see text]-[Formula: see text]-permutable subgroups on the structure of finite groups.
A group is called a CA-group if the centralizer of every non-central element is abelian. Furthermore, a group is called a minimal non-CA-group if it is not a CA-group itself, …
A group is called a CA-group if the centralizer of every non-central element is abelian. Furthermore, a group is called a minimal non-CA-group if it is not a CA-group itself, but all of its proper subgroups are. In this paper, we give a classification of the finite non-solvable minimal non-CA-groups.
Let G be a finite group, σ={σi|i∈I} a partition of the set of all primes P and σ(G)={σi|σi∩π(|G|)≠∅}. A set H of subgroups of G is said to be a …
Let G be a finite group, σ={σi|i∈I} a partition of the set of all primes P and σ(G)={σi|σi∩π(|G|)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i∈I and H contains exactly one Hall σi-subgroup of G for every σi∈σ(G). G is said to be σ-full if G possesses a complete Hall σ-set. We say a subgroup H of G is sσ-quasinormal (supplement-σ-quasinormal) in G if there exists a σ-full subgroup T of G such that G = HT and H permutes with every Hall σi-subgroup of T for all σi∈σ(T). In this article, we obtain some results about the sσ-quasinormal subgroups and use them to determine the structure of finite groups. In particular, some new criteria of p-nilpotency, solubility, supersolubility of a group are obtained.
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 <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> …
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 <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 number; a character 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> is called <italic><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>-constant</italic> if it takes a constant value on all the elements 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> whose order is divisible by <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>. This is a generalization of the very important concept of characters 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>-defect zero. In this paper, we characterize the finite <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 groups having a faithful irreducible character that 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>-constant and not 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>-defect zero, and we will show that a non-<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 group with this property is an almost-simple group.
Let G be a nontrivial transitive permutation group on a finite set [Formula: see text] and recall that an element of G is a derangement if it has no fixed …
Let G be a nontrivial transitive permutation group on a finite set [Formula: see text] and recall that an element of G is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called elusive groups that do not contain any derangements of prime order. In a recent paper, Burness and the author introduced the family of almost elusive groups, which contain a unique conjugacy class of derangements of prime order. In this paper, we complete the classification of the quasiprimitive almost elusive groups.
Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, b D ) be a Sylow B-subgroup of S(n). …
Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, b D ) be a Sylow B-subgroup of S(n). We consider the fusion system [Formula: see text] and determine a precise formula for its essential rank. In addition, the p-blocks B which admit a p-local subgroup of S(n) controlling B-fusion are characterized.
Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two …
Let G be a non-abelian group. The non-commuting graph [Formula: see text] of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with [Formula: see text], where 57 is the clique number of the non-commuting graph of the projective special linear group PSL (2,7). We also determine [Formula: see text] for all finite minimal simple groups G.
Abstract We call a subgroup H of a group G nearly s-normal in G if there exists N ⊴ G such that HN ⊴ G and H ∩ N ≤ …
Abstract We call a subgroup H of a group G nearly s-normal in G if there exists N ⊴ G such that HN ⊴ G and H ∩ N ≤ H sG , where H sG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups. Key Words: Finite groupsNearly s-normal subgroups p-Nilpotent groupsSupersoluble groups2000 Mathematics Subject Classification: 20D1020D2020D25 ACKNOWLEDGMENT Research is supported by NNSF of China (Grant #11071229) and by Postgraduate Key Fund of Xuzhou Normal University.
This thesis concerns the study of homogeneous factorisations of complete graphs with edge-transitive factors. A factorisation of a complete graph $K_n$ is a partition of its edges into disjoint classes. …
This thesis concerns the study of homogeneous factorisations of complete graphs with edge-transitive factors. A factorisation of a complete graph $K_n$ is a partition of its edges into disjoint classes. Each class of edges in a factorisation of $K_n$ corresponds to a spanning subgraph called a factor. If all the factors are isomorphic to one another, then a factorisation of $K_n$ is called an isomorphic factorisation. A homogeneous factorisation of a complete graph is an isomorphic factorisation where there exists a group $G$ which permutes the factors transitively, and a normal subgroup $M$ of $G$ such that each factor is $M$-vertex-transitive. If $M$ also acts edge-transitively on each factor, then a homogeneous factorisation of $K_n$ is called an edge-transitive homogeneous factorisation. The aim of this thesis is to study edge-transitive homogeneous factorisations of $K_n$. We achieve a nearly complete explicit classification except for the case where $G$ is an affine 2-homogeneous group of the form $Z_p^R \rtimes G_0$, where $G_0 \leq ΓL(1,p^R)$. In this case, we obtain necessary and sufficient arithmetic conditions on certain parameters for such factorisations to exist, and give a generic construction that specifies the homogeneous factorisation completely, given that the conditions on the parameters hold. Moreover, we give two constructions of infinite families of examples where we specify the parameters explicitly. In the second infinite family, the arc-transitive factors are generalisations of certain arc-transitive, self-complementary graphs constructed by Peisert in 2001.
Let G be a finite group and H a subgroup of G. We say that H is an ℋ-subgroup in G if NG(H) ∩ Hg ≤ H for all g …
Let G be a finite group and H a subgroup of G. We say that H is an ℋ-subgroup in G if NG(H) ∩ Hg ≤ H for all g ∈ G; H is called weakly ℋ-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ℋ-subgroup in G. We say that H is weakly ℋ -embedded in G if G has a normal subgroup K such that HG = HK and H ∩ K is an ℋ-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ℋ-embedded in G. Our results improve and generalize several recent results in the literature.
We complete the proof of the height conjecture for <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 groups, using the classification of finite simple groups.
We complete the proof of the height conjecture for <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 groups, using the classification of finite simple groups.
We show that if G is a group of permutations on a set of n points and if \G/G'\ denotes the order of its largest abelian quotient, then either \G/G'\ …
We show that if G is a group of permutations on a set of n points and if \G/G'\ denotes the order of its largest abelian quotient, then either \G/G'\ = 1 or there is a prime p dividing \G/G'\ such that \G/G'\ < p n/p .Equality holds if and only if G is a p-group which is the direct product of its transitive constituents, with each of those having order /?, except when p = 2 in which case one must also allow as transitive constituents the groups of order 4, the dihedral group of order 8 and degree 4, and the extraspecial group of order 32 and degree 8.
Let σ be some partition of the set of all primes and H a complete Hall σ-set of a finite group G. A subgroup H of G is said to …
Let σ be some partition of the set of all primes and H a complete Hall σ-set of a finite group G. A subgroup H of G is said to be σ-conditionally permutable in G if for any subgroup A∈H, there exists an element x∈G such that HAx=AxH. In this article, we investigate the influence of σ-conditionally permutable subgroups on the structure of finite groups.
The Baer--Suzuki theorem says that if $p$ is a prime, $x$ is a $p$-element in a finite group $G$ and $\langle x, x^g \rangle$ is a $p$-group for all $g …
The Baer--Suzuki theorem says that if $p$ is a prime, $x$ is a $p$-element in a finite group $G$ and $\langle x, x^g \rangle$ is a $p$-group for all $g \in G$, then the normal closure of $x$ in $G$ is a $p$-group. We consider the case where $x^g$ is replaced by $y^g$ for some other $p$-element $y$. While the analog of Baer--Suzuki is not true, we show that some variation is. We also answer a closely related question of Pavel Shumyatsky on commutators of conjugacy classes of $p$-elements.
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔉</m:mi> </m:math> {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℒ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> …
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔉</m:mi> </m:math> {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℒ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}(G)} be the lattice of all subgroups of G . A chief <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> {H/K} factor of G is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔉</m:mi> </m:math> {\mathfrak{F}} -central in G if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⋊</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>G</m:mi> </m:msub> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∈</m:mo> <m:mi>𝔉</m:mi> </m:mrow> </m:math> {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mrow> <m:mi>c</m:mi> <m:mo></m:mo> <m:mi>𝔉</m:mi> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> {H/K} of G between <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi>G</m:mi> </m:msub> </m:math> {A_{G}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>A</m:mi> <m:mi>G</m:mi> </m:msup> </m:math> {A^{G}} is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔉</m:mi> </m:math> {\mathfrak{F}} -central in G ; <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>𝔉</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>A</m:mi> <m:mi>G</m:mi> </m:msup> <m:mo>/</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>G</m:mi> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:mi>𝔉</m:mi> </m:mrow> </m:math> {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mrow> <m:mi>c</m:mi> <m:mo></m:mo> <m:mi>𝔉</m:mi> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔉</m:mi> </m:math> {\mathfrak{F}} is a Fitting formation, the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>𝔉</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℒ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}(G)} . We also study conditions under which the lattice <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mrow> <m:mi>c</m:mi> <m:mo></m:mo> <m:mi>𝔑</m:mi> </m:mrow> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.
In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.
In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.
Abstract We present a direct combinatorial proof of the characterization of the degree of transivity of a finite permutation group in terms of the Bell numbers. Subject classification ( Amer. …
Abstract We present a direct combinatorial proof of the characterization of the degree of transivity of a finite permutation group in terms of the Bell numbers. Subject classification ( Amer. Math.Soc. ( MOS ) 1970 ): 20 B 20.