On a Class of Doubly Transitive Groups: II

Authors

Michio Suzuki

Type: Article

Publication Date: 1964-05-01

Citations: 726

DOI: https://doi.org/10.2307/1970408

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Abstract

Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of the investigation is to classify the groups which do not contain a regular normal subgroup of order 1 + N in case N is even. (Author)

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Annals of Mathematics
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On a class of doubly transitive groups

1972-09-01

Ernest E. Shult

The purpose of this paper is to prove the following theorem" THEOREM.Let G be a transitive group of permutations on the (finite) set of letters .Let G be the subgroup … The purpose of this paper is to prove the following theorem" THEOREM.Let G be a transitive group of permutations on the (finite) set of letters .Let G be the subgroup of G fixing the letter a in .Suppose G contains a normal subgroup Q of even order, which is regular on (a).Then either a G is a subgroup of the group of semi-linear transformations over a near field of odd characteristic or (b) G is an extension of one of the groups SL(2, q), Sz(q) or U(3, q) by a subgroup of its outer automorphism group.(I 1 Aq, 1 "4-q or 1 -b q8 in these three respective cases (q 2).)Essentially "half" of this theorem was proved by Suzuki [8], under the assumption that the quotient group G/Q had odd order.We therefore consider only the case that G/Q has even order.Since Q is regular on t2 (a), we may express G as a semidirect product G Q where G G n G, the subgroup of permutations fixing both a and f.For the rest of this paper, all groups considered are finite.We write IX[ for the cardinality of set X.If X is a subset of a group G, we write X c_ G, and if X is a subgroup of G, we write X _< G.If X _ _ _ G, (X) will denote the subgroup of G generated by X.If X is a subset of G, X denotes the set of all conjugate sets {glXgig G}.We will frequently write (X) instead of the more cumbersome ((Jr, xa Y).This is the normal closure of X in G and represents the smallest normal subgroup of G containing X.If M is a group of (right) operators of a group G it will frequently be convenient to proceed with computations in the semi-direct product GM and also to view GM as a group of right operators of G, the elements of G acting by conjugation.Ac- tion of these operators is indicated by exponential notation.Thus if x e G, g-lxg may be written x g and if is an automorphism of G, we may writeThe commutator x-y-xy is written [x, y].If a is an automorphism of G and if x e G, then the commutator Ix, ] is assumed to be computed in the semidirect product G(a), so Ix, r] x-.x.If r is a set of primes, a v-group is a group whose order involves only primes in r. /ks usual, r' denotes the complement of r in the set of all primes.If r consists of a single prime p, the symbol p (rather than {p} may replace the symbol r in the notation of

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On a class of doubly transitive permutation groups

1960-06-01

Walter Feit

IntroductionIn this pper we will study permutation groups stisfying the following conditions.HvPOWHESS I. G is a doubly transitive permutation group on m 1 letters in which no nontrivial permutation leaves … IntroductionIn this pper we will study permutation groups stisfying the following conditions.HvPOWHESS I. G is a doubly transitive permutation group on m 1 letters in which no nontrivial permutation leaves three letters fixed.All known examples of permutation groups G stisfying Hypothesis I either contain normal subgroup of order m -1 or re contained in n exactly triply transitive permutation group Go, with [G0"G] _-< 2. In the ltter cse it is known [10] that m p for some prime p nd that the Sylow p-groups of G are belin.In view of this it seems reasonable to coniecture that the only permutation groups stisfying Hypothesis I re the ones just mentioned.In this pper we prove the following result which is step in the direction of the conjecture.THEOREM 1.Let G be a permutation group of order qm(m -1) which satis- fies Hypothesis I. Then either G contains a normal subgroup of order m 1, or m p for some prime p.In the latter case, [S" S] < 4q , where S is the Sylow p-group of G, and if S 11}, there exists an exactly triply transitive per- mutation group Go containing G such that [Go" G] _-< 2.Section 2 is devoted to the proof of Theorem 2 which is the min result of this pper.This theorem enables one to compute lrge prt of the char- acter table for groups G which contain subgroup M stisfying certain condi- tions (Hypothesis II in Section 2).The proof of Theorem 2 uses the fund- mental result recently proved by J. G. Thompson [9], which together with the results of [5] nd [7] show that the regular subgroup of Frobenius group is nilpotent.The special cse of Theorem 2 in which the subgroup M is belin ws proved by R. Bruer nd M. Suzuki [8] nd hs turned out to be power- ful tool in the study of finite linear groups (see for example [3], [8]).Since

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On Some Doubly Transitive Permutation Groups of Degree N And Order 6n(n – 1)

1970-03-01

Shiro Iwasaki , Hiroshi Kimura

Let Ω be the set of symbols 1,2,..., n. Let be a doubly transitive group on Ω of order 6n(n — 1) not containing a regular normal subgroup and let … Let Ω be the set of symbols 1,2,..., n. Let be a doubly transitive group on Ω of order 6n(n — 1) not containing a regular normal subgroup and let be the stabilizer of the set of symbols 1 and 2.

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On a Class of Doubly Transitive Groups

1961-08-01

J. L. Zemmer

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On a Class of Doubly Transitive Groups

1962-01-01

Michio Suzuki

On the class of doubly transitive groups

1914-01-01

W. A. Manning

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On a class of doubly transitive permutation groups

1969-06-01

George Glauberman

Let G be a finite permutation group.We say that G is a Zassenhaus group if G is doubly transitive and if no non-identity element of G leaves three or more … Let G be a finite permutation group.We say that G is a Zassenhaus group if G is doubly transitive and if no non-identity element of G leaves three or more symbols fixed.The Zassenhaus groups have been determined by Zassenhaus [7, 8], Feit [3], Suzuki [6], and Ito [5].In this paper we present an alternate proof of Ito's result.THEOREM (Ito).Let G be a Zassenhaus group of degree m -k 1 that does not contain a regular normal subgroup.If m is a power of an odd prime p, then G has an Abelian Sylow p-subgroup.Our proof uses the notation of Feit [3].Let N be the subgroup of G fixing one symbol, and let Q be the subgroup of G fixing an additional symbol.Let g G] and q ]q I. Since G has no regular normal subgroup, G is not Frobenius group, and q > 1.Thus N acts as a Frobenius group on the sym- bols it moves.Let M be the regular normal subgroup of N. Thus M m, and N MQ, M n Q 1, INI mq, g (m / 1)INI (m + 1)mq.

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On a class of doubly, but not triply transitive permutation groups

1967-12-01

Noboru Ιτο

On some doubly transitive permutation groups of degree $n$ and order $2^{l}(n-1)n$

1970-07-01

Hiroshi Kimurâ

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On the Sylow subgroups of a doubly transitive permutation group. II

1975-06-01

Cheryl E. Praeger

Some class of doubly transitive groups of degree $n$ and order $4q(n-1)n$ where $q$ is an odd number

1970-07-01

Hiroshi Kimura , Hiroyoshi Yamaki

Some class of doubly transitive groups of degree $n$ and order $4q(n-1)n$ where $q$ is an odd number Some class of doubly transitive groups of degree $n$ and order $4q(n-1)n$ where $q$ is an odd number

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On a class of doubly transitive permutation groups of degree $pq+1$

1977-06-01

David Chillag

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Abstract groups as doubly transitive permutation groups

1972-01-01

Russell Merris

T he question considered is this: Which abstract groups have represe ntations as doubly transitive permutation groups?Moreover, given an abstrac t group, can all doubly transitive represen tations be found?T … T he question considered is this: Which abstract groups have represe ntations as doubly transitive permutation groups?Moreover, given an abstrac t group, can all doubly transitive represen tations be found?T he paper is expository.Various results which bear on the question are presented in a n elemen• tary way.

Cyclic permutations in doubly-transitive groups

1997-01-01

John P. McSorley

(1997). Cyclic permutations in doubly-transitive groups. Communications in Algebra: Vol. 25, No. 1, pp. 33-35. (1997). Cyclic permutations in doubly-transitive groups. Communications in Algebra: Vol. 25, No. 1, pp. 33-35.

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On doubly transitive permutation groups of degree prime squared plus one

1978-11-01

Peter J‎. Cameron

Abstract Groups with the property of the title were considered by Chillag (1977); this paper completes his results by showing that, with known exceptions, they are triply transitive. Abstract Groups with the property of the title were considered by Chillag (1977); this paper completes his results by showing that, with known exceptions, they are triply transitive.

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On a class of doubly transitive permutation groups

1962-06-01

Noboru Ιτο

NOBORULet 2 be the set of symbols l, m + i.Let @ be a doubly transitive permutation group on 2 in which no nontrivial permutation leaves three sym- bols fixed.Such … NOBORULet 2 be the set of symbols l, m + i.Let @ be a doubly transitive permutation group on 2 in which no nontrivial permutation leaves three sym- bols fixed.Such a group (9 will be called a Zassenhaus group.On the structure of Zassenhaus groups Feit [4] proved recently the following elegant theorem: Let @ be a Zassenhaus group of degree m + i, which contains no normal subgroup of order m -I.Then m must be a power of a prime number: m pe.Let 9)} be a Sylow p-subgroup of @, and let J' be the commutator subgroup of )L Then the index of ' in must be smaller than 4q2, where q is the order of the subgroup , which consists of all the permutations leaving each of the symbols I and 2 fixed.Moreover if )} is abelian, then q -> (m i)/2.Now the purpose of this paper is to prove the following.THEOREM.If m is odd, then must be abelian.1.In the following (9 denotes always a Zassenhaus group of even degree m + 1, which contains no normal subgroup oforderm + 1.Let F (i 0, 1,2) be the set of all the permutations in @, each of which fixes just i symbols of .Then according to our assumptions on @ we obtain the following decomposi- tion of @ into its mutually disjoint subsets" (9 F0 -{-F1 -}-F2 + 1}, where 1 is the identity element of @.Since (9 is doubly transitive, (9 possesses an irreducible character B, whose values can be written as follows"(1) m for X= 1, 1 for X B(X) 0 for --1 for XeF0.. Let @1 be the subgroup of (9, which consists of all the permutations leaving the symbol 1 fixed.Then we can choose an )} in the theorem of Felt in the following way: )} is a normal subgroup of @1 and satisfies the condi- tions that @1 )?: and n : 1.Now we assume that (2.1) is not abelian.Therefore the purpose of our proof is to derive a contradiction from this

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On automorphisms of doubly transitive permutation groups [55]

2012-03-24

A note on doubly transitive groups

1966-11-01

Peter Lorimer

W. Feit [1], N. Itô [2] and M. Suzuki [3] have determined all doubly transitive groups with the property that only the identity fixes three symbols. It is of interest … W. Feit [1], N. Itô [2] and M. Suzuki [3] have determined all doubly transitive groups with the property that only the identity fixes three symbols. It is of interest to the theory of projective planes to determine whether any of these groups contain a sharply doubly transitive subset (see Definition 1). It is found that if such a group G contains such a subset R then R is a normal subgroup of G , i.e. R is a doubly transitive normal subgroup of G in which only the identity fixes two symbols.

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On the order of doubly transitive permutation groups

1982-10-01

László Babai

A Characterization of the Simple Group U3(5)

1970-03-01

Koichiro Harada

In this note we consider a finite group G which satisfies the following conditions: (0. 1) G is a doubly transitive permutation group on a set Ω of m + … In this note we consider a finite group G which satisfies the following conditions: (0. 1) G is a doubly transitive permutation group on a set Ω of m + 1 letters, where m is an odd integer ≥ 3, (0. 2) if H is a subgroup of G and contains all the elements of G which fix two different letters α, β , then H contains unique permutation h 0 ≠ 1 which fixes at least three letters, (0. 3) every involution of G fixes at least three letters, (0. 4) G is not isomorphic to one of the groups of Ree type.

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最近の日本の数学(そのIII) 有限群論

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敬義河田 , 英一坂内

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Nick Gill

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Carlisle S. H. King

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Nikolai Gordeev , Fritz Grunewald , Boris Kunyavskiı̆ +1 more

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John K. McVey

Strongly Real Beauville Groups III

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Damiano Rossi , Benjamin Sambale

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2001-04-01

Charles W. Eaton

Finite groups and subgroup-permutability

1995-12-01

Mariagrazia Bianchi , Anna Gillio Berta Mauri , Libero Verardi

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Martin W. Liebeck , Cheryl E. Praeger , Jan Saxl

Subgroups of finite Chevalley groups

1986-02-28

А. С. Кондратьев

CONTENTS Introduction § 1. Primary subgroups § 2. Local subgroups § 3. Conjugacy classes and centralizers of elements § 4. Embeddings § 5. Intermediate subgroups § 6. Permutation representations § … CONTENTS Introduction § 1. Primary subgroups § 2. Local subgroups § 3. Conjugacy classes and centralizers of elements § 4. Embeddings § 5. Intermediate subgroups § 6. Permutation representations § 7. Maximal subgroups § 8. Other subgroups References

Groups containing a strongly embedded subgroup

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Д. В. Лыткина , V. D. Mazurov

Groups whose elements are not conjugate to their powers

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Andreas Bächle , Benjamin Sambale

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On a property like Sylow's of some finite groups

1980-12-01

Zvi Arad , David Chillag

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>designs admitting automorphism groups with socle Sz(q)

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Shangzhao Li , Weijun Liu , Xianhua Li

Infinite locally finite groups of type $PSL (2,K)$ or $Sz(K)$ are not minimal under certain conditions

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Javier Otal , J.M. Peña

In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration .We obtain some of such criteria as a consequence of … In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration .We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simple groups .

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Some results on 2-transitive groups

1971-03-01

William M. Kantor , Gary M. Seitz

Nonabelian composition factors of a finite group whose all maximal subgroups are hall

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Н. В. Маслова

Twisted tensor product subgroups of finite classical groups

1999-01-01

Mark E. Schaffer

If Cln(qr) denotes a classical group with natural module W of dimensioa n over Fqr , then the twisted tensor product module is realised over Fq, and yields an embedding … If Cln(qr) denotes a classical group with natural module W of dimensioa n over Fqr , then the twisted tensor product module is realised over Fq, and yields an embedding . These embeddings play a significant role in the subgroup structure of classical groups; for example, Seitz [18] shows that any maximal absolutely irreducible subgroup defined over a proper extension field of Fq is of this form. In this paper we study the precise nature of these embeddings, and go on to investigate their maximality or otherwise. We show that the normaliser of Cln(qr) is usually maximal, with an explicit list of just 4 families of exceptions.

The probabilistic zeta function of PSL(2,q), of the Suzuki groups ²B2(q) and of the Ree groups ²G2(q)

2009-03-02

Massimiliano Patassini

We study the Dirichlet polynomial P G (s) of the groups G = PSL(2, q), 2 B 2 (q), and 2 G 2 (q).For such G we show that if … We study the Dirichlet polynomial P G (s) of the groups G = PSL(2, q), 2 B 2 (q), and 2 G 2 (q).For such G we show that if H is a group satisfying P H (s) = P G (s), then H/Frat(H) ∼ = G.We also prove that, when q is not a prime number, P G (s) is irreducible in the ring of Dirichlet polynomials.Finally, we prove that the coset poset of G is noncontractible.

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Suzuki-invariant codes from the Suzuki curve

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Abdulla Eid , Hilaf Hasson , Amy Ksir +1 more

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E. P. Vdovin

A Characteristic Property of A8

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Wujie Shi

Zur Vermutung von Sims �ber primitive Permutationsgruppen II

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Michael B�rker , Wolfgang Knapp

Automorphism groups of edge-transitive maps

2016-01-01

Gareth A. Jones

For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a … For each of the 14 classes of edge-transitive maps described by Graver and Watkins, necessary and sufficient conditions are given for a group to be the automorphism group of a map, or of an orientable map without boundary, in that class. Extending earlier results of Siran, Tucker and Watkins, these are used to determine which symmetric groups $S_n$ can arise in this way for each class. Similar results are obtained for all finite simple groups, building on work of Leemans and Liebeck, Nuzhin and others on generating sets for such groups. It is also shown that each edge-transitive class realises finite groups of every sufficiently large nilpotence class or derived length, and also realises uncountably many non-isomorphic infinite groups. Edge-transitive embeddings of complete graphs are classified, and there is a detailed discussion of edge-transitive maps with boundary.

Brauer’s height conjecture for 𝑝-solvable groups

1984-01-01

David Gluck , Thomas R. Wolf

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.

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Generation of exceptional groups of Lie-type

1992-01-01

ThomasS. Weigel

Character tables of the maximal parabolic subgroups of the Ree groups 2F4(q2)

2010-04-01

Frank Himstedt , Shih-Chang Huang

Abstract We compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of the simple Ree groups 2 F 4 ( q 2 ). For … Abstract We compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of the simple Ree groups 2 F 4 ( q 2 ). For one of the maximal parabolic subgroups, we find an irreducible character of the unipotent radical that does not extend to its inertia subgroup.

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Flag-transitive symmetric designs

2003-01-01

Eugenia O’Reilly-Regueiro

Perfect commuting graphs

2016-06-07

John R. Britnell , Nick Gill

Abstract We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect. Abstract We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.

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Finite groups whose solvable subgroups have 2-length equal to 1

1972-07-01

V. D. Mazurov

Neighbour-transitive codes in Johnson graphs

2013-11-01

Robert A. Liebler , Cheryl E. Praeger

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of … The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such that the subgroup G of graph automorphisms leaving \Gamma invariant is transitive on both the set \Gamma of `codewords' and also the set of `neighbours' of \Gamma, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group on V and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G lies in Sym(V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3 in J(v,k), then G is 2-transitive on V. We examine many of the infinite families of finite 2-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.

Combinatorial categories and permutation groups

2013-09-24

Gareth A. Jones

The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic … The regular objects in various categories, such as maps, hypermaps or covering spaces, can be identified with the normal subgroups N of a given group \Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to enumerate such objects with a given finite automorphism group G, how to represent them all as quotients of a single regular object U(G), and how they are acted on by the outer automorphism group of \Gamma. Examples constructed include kaleidoscopic maps with trinity symmetry.

Characters of Positive Height in Blocks of Finite Quasi-Simple Groups

2014-10-15

Olivier Brunat , Gunter Malle

Journal Article Characters of Positive Height in Blocks of Finite Quasi-Simple Groups Get access Olivier Brunat, Olivier Brunat 1Université Paris-Diderot Paris 7, Institut de Mathématiques de Jussieu, UFR de Mathé-matiques, … Journal Article Characters of Positive Height in Blocks of Finite Quasi-Simple Groups Get access Olivier Brunat, Olivier Brunat 1Université Paris-Diderot Paris 7, Institut de Mathématiques de Jussieu, UFR de Mathé-matiques, Case 7012, 75205 Paris Cedex 13, France Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Gunter Malle Gunter Malle 2FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2015, Issue 17, 2015, Pages 7763–7786, https://doi.org/10.1093/imrn/rnu171 Published: 15 October 2014 Article history Received: 15 May 2014 Revision received: 12 August 2014 Accepted: 09 September 2014 Published: 15 October 2014

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On infinite groups with a given strongly isolated 2-subgroup

2000-08-01

A. I. Sozutov , Н. М. Сучков

Algorithmic problems in twisted groups of Lie type

2008-01-01

Henrik Bäärnhielm

This thesis contains a collection of algorithms for working with the twisted groups of Lie type known as Suzuki groups, and small and large Ree groups. The two main problems … This thesis contains a collection of algorithms for working with the twisted groups of Lie type known as Suzuki groups, and small and large Ree groups. The two main problems under consideration are constructive recognition and constructive membership testing. We also consider problems of generating and conjugating Sylow and maximal subgroups. The algorithms are motivated by, and form a part of, the Matrix Group Recognition Project. Obtaining both theoretically and practically efficient algorithms has been a central goal. The algorithms have been developed with, and implemented in, the computer algebra system MAGMA.

Finite groups in which every irreducible character vanishes on at most two conjugacy classes

2000-01-01

Mariagrazia Bianchi , David Chillag , Anna Gillio

Partial linear spaces with a rank 3 affine primitive group of automorphisms

2021-05-05

John Bamberg , Alice Devillers , Joanna B. Fawcett +1 more

A partial linear space is a pair (P, L) where P is a non-empty set of points and L is a collection of subsets of P called lines such that … A partial linear space is a pair (P, L) where P is a non-empty set of points and L is a collection of subsets of P called lines such that any two distinct points are contained in at most one line, and every line contains at least two points.A partial linear space is proper when it is not a linear space or a graph.A group of automorphisms G of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when G is transitive of rank 3 on points.In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of AΓL1(q).Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups.We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.

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A property of Suzuki groups

1972-09-01

В. М. Левчук

On Characterizations of Linear Groups III

1962-12-01

Michio Suzuki

In his address at the International Congress of Mathematicians at Amsterdam [1] Professor R. Brauer proposed a problem of characterizing various groups of even order by the properties of the … In his address at the International Congress of Mathematicians at Amsterdam [1] Professor R. Brauer proposed a problem of characterizing various groups of even order by the properties of the involutions contained in these groups and he gave characterizations of the general projective linear groups of low dimensions along these lines. The detail of the one-dimensional case has been published in [5], but the two-dimensional case has not appeared yet in detail. His work was followed by Suzuki [7], Feit [6] and Walter [11]. The present paper is a continuation of [73 and discusses a characterization of the two-dimensional projective unitary group over a finite field of characteristic 2. The precise conditions which characterize the group in question will be stated in the first section. The method employed here is similar to the one used in [7]. An application of group characters provides a formula for the order. However a difficulty comes in when one attempts to identify the group. In order to overcome this difficulty we will use a method primarily designed to study a class of doubly transitive permutation groups (cf. [9]). We need also a group theoretical characterization of a class of doubly transitive groups called ( ZT )-groups. This is a generalization of a result in [8], and may be of in dependent interest.

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On generalized conjugate classes in a finite group

1959-09-01

Rimhak Ree

Throughout this note G will denote a finite group, and a fixed homomor- phism of G into itself.may be an automorphism of G. Two elements a and b in G … Throughout this note G will denote a finite group, and a fixed homomor- phism of G into itself.may be an automorphism of G. Two elements a and b in G will be called -conjugate if there exists an element x in G such that a x bx.This is an equivalence relation, and all elements in G are par- titioned into -classes.If is the identity automorphism, then -conjugacy reduces to the "ordinary" coniugacy in groups.A subset S of G will be called -invariant if x e S implies x e S. In this note we shall prove the fol- lowing" THEOREM 1.The number of -classes equals the number of -invariant classes of conjugate elements in G.An interesting feature of the above theorem is that, although the theorem itself does not involve group characters, it does not seem to be proved easily without using group characters.The author has been unable to obtain such a proof.Actually Theorem 1 is an immediate consequence of the following two theorems.THEOREM 2. The number o] -classes in G is equal to the number o] -in- variant irreducible ordinary characters of G. THEOREM 3. Let p be an arbitrary prime number.Then the number ofinvariant irreducible modular characters (with respect to p) is equal to the num- ber o] -invariant p-regular classes o] conjugate elements in G.In particular, the number o] -invariant ordinary characters is equal to the number of -invariant classes o] conjugate elements in G.Here, a function (x) defined on a -invariant subset S of G is called -in- variant if (x) (x) for all x e S; a class of conjugate elements is called p-regular if it consists of elements of order prime to p. Theorem 2 above is a generalization of a result of Ado [1], who proved Theorem 2 for the case where is an automorphism.In his proof Ado made use of the inverse mapping -1.The method we use here to prove Theorem 2 is a rather trivial modification of Ado's.Proof of Theorem 2. Let xl, x be all irreducible ordinary characters

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Theory of Groups of Finite Order

2012-06-07

W. Burnside

The British mathematician William Burnside (1852–1927) and Ferdinand Georg Frobenius (1849–1917), Professor at Zurich and Berlin universities, are considered to be the founders of the modern theory of finite groups. … The British mathematician William Burnside (1852–1927) and Ferdinand Georg Frobenius (1849–1917), Professor at Zurich and Berlin universities, are considered to be the founders of the modern theory of finite groups. Not only did Burnside prove many important theorems, but he also laid down lines of research for the next hundred years: two Fields Medals have been awarded for work on problems suggested by him. The Theory of Groups of Finite Order, originally published in 1897, was the first major textbook on the subject. The 1911 second edition (reissued here) contains an account of Frobenius's character theory, and remained the standard reference for many years.

On a class of doubly transitive permutation groups

1960-06-01

Walter Feit

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Finite groups with nilpotent centralizers

1961-01-01

Michio Suzuki

Introduction.The purpose of this paper is to clarify the structure of finite groups satisfying the following condition:(CN) : the centralizer of any nonidentity element is nilpotent.Throughout this investigation we consider … Introduction.The purpose of this paper is to clarify the structure of finite groups satisfying the following condition:(CN) : the centralizer of any nonidentity element is nilpotent.Throughout this investigation we consider only groups of finite order.A group is called a (P)-group if it satisfies a group theoretical property (P).In this paper we shall clarify the structure of nonsolvable (CN)-groups and classify them as far as possible.This goal has been attained in a sense which we shall explain later.If we replace in (CN) the assumption of nilpotency by being abelian we get a stronger condition (CA).The structure of (CA)-groups has been known.In fact after an initial attempt by K. A. Fowler in his thesis [8], Wall and the author have shown that a nonsolvable (CA)-group of even order is isomorphic with LF(2, q) lor some q = 2n>2.A few years later the author [12] has succeeded in proving a particular case of Burnside's conjecture for (CA)groups, namely a nonsolvable (CA)-group has an even order.Quite recently Feit, M. Hall and Thompson [7] have proved the Burnside's conjecture for (CN)-groups.We can therefore consider groups of even order and focus our attention to the centralizers of involutions.We consider the condition (CIT):(CIT) : a group is of even order and the centralizer of any involution is a 2-group.There is no apparent connection between the class of (CN)-groups and the class of (CIT)-groups.But a nonsolvable (CN)-group is a (CIT)-group (Theorem 4 in Part I).This theorem reduces the study of nonsolvable (CN)groups to that of (CIT)-groups.Both properties (CN) and (CIT) are obviously hereditary to subgroups (provided that we consider only subgroups of even order in the case of (CIT)).Although it is true that a homomorphic image of a (CN)-group is also a (CN)-group (this statement is false for infinite groups), it is not an obvious statement.On the other hand it is not difficult to show that a factor group of a (CIT)-group is a (CIT)-group, provided that the order is even.This is due to the following characterization of (CIT)-groups : namely a (CIT)-group is a group of even order containing no element of order 2p with p>2 and vice versa.This makes the study of (CIT)-groups somewhat easier.The large part of this paper concerns the structure of (CIT)groups.

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FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER

1959-04-01

John Thompson

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Zweifach transitive, aufl�sbare Permutationsgruppen

1957-12-01

Bertram Huppert

ON FINITE GROUPS OF EVEN ORDER WHOSE 2-SYLOW GROUP IS A QUATERNION GROUP

1959-12-01

Richard Brauer , Michio Suzuki

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Suzuki $2$-groups

1963-03-01

Graham Higman

GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, … GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, except that we do not exclude the possibility that two or more of the groups that we list may be isomorphic.The investigation is perhaps not without interest simply as an example of the use of linear methods in p-group theory; but the main motivation for it is that some result along these lines is needed by Suzuki in his classification [4] of ZT-groups.It is a pleasure to acknowledge that this paper is, in a direct way, a fruit of the special year in Group Theory organized by the Department of Mathematics at the University of Chicago.A 2-group with only one involution, that is, a eyelie or generalised quaternion group obviously has the property under discussion; and an abelian group has it if and only if it is a direct product of eyelie 2-groups all of the same order.It is convenient to exclude these eases from the beginning, and define a Suzulci 2-group as a non-abelian 2-group with more than one involution, having a eyelie group of automorphisms which permutes its involutions transi- tively.Evidently, the involutions of a Suzuki 2-group G all belong to its center, and so constitute, with the identity, an elementary abelian subgroup fh(G) of order q 2", n > 1.We shall show that fI(G) Z(G) q(G) G', so that G is of exponent 4 and class 2. The automorphism ( which permutes cyclically the q 1 involutions evidently has order divisible by q 1.We shall show that can be taken to have order precisely q 1, and so to be regular.The order of G is either q or qa.In many ways, it would be more satisfactory to impose on G the simpler, weaker condition that the involutions of G are permuted transitively by the full automorphism group of G. Possibly such a relaxation would not bring in any large class of new groups; but the condition seems to be very hard to handle.However, a little of our argument extends to the general ease, and this part has been stated for that ease.The methods used are similar to those involving the associated Lie ring (el.e.g. [2]),but we shall not construct this ring explicitly.The setup, which we shall presuppose, is as follows.If H is a subgroup of the 2-group G, and K a normal subgroup of H with elementary abelian factor group H/K,

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Finite Groups Admitting a Fixed-Point-Free Automorphism of Order 4

1961-01-01

Daniel Gorenstein , I. N. Herstein

Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group … Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group admitting a fixed-point-free automorphism (i. e., leaving only the identity element fixed) of prime order must be nilpotent. However, non-nilpotent groups are known which admit fixedpoint-free automorphisms of composite order. In all these cases one notices that the groups in question are solvable. Although the sample is rather restricted, it is not too unnatural to ask whether the condition that a finite group admit such an automorphism is strong enough to force solvability of the group. This question is related to another problem, which seems. equally difficult, which asks whether a finite group containing a cyclic subgroup which is its own normalizer must be composite. In the present paper we shall prove that a group G possessiilg a fixedpoint-f ree automorphism of order 4 is solvable. Although many of the ideas used carry over to the case in which 4 has order pq, and especially 2q, our key lemmas use the fact that 4 has order 4 in a crucial way. The proof depends upon a theorem of Philip Ilall which asserts that a finite group G is solvable if for every factorization of o(G) into relatively prime numbers m and n, G contains a subgroup of order m. We show (Lemma 7) that a group G which has a fixed-point-free automorphism of order 4 satisfies the conditions of H all's theorem. Once we k-now that G is solvable it is not difficult to prove that its commutator subgroup is nilpotent (Theorem 2). This -fact was also observed by Thompson. Graham Higman has shown [3] that there is a bound to the class of a p-group P which possesses an automorphism p of prime order q without fixedpoints. This does not carry over to automorphisms of composite order, for at the end of the paper we give an example due to Thompson of a family of p-groups of arbitrary high class each of which admits a fixed-point-free automorphism of order 4.

A Characterization of the Simple Groups SL(2,2 a )

1960-04-01

Walter Feit

A Family of Simple Groups Associated with the Simple Lie Algebra of Type (F 4 )

1961-07-01

Rimhak Ree