Abstract In 1878, Jordan showed that a finite subgroup of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo form="prefix">GL</m:mo> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>ℂ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> ${\operatorname{GL}(n,\mathbb {C})}$ must possess an abelian normal subgroup …
Abstract In 1878, Jordan showed that a finite subgroup of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo form="prefix">GL</m:mo> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>ℂ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> ${\operatorname{GL}(n,\mathbb {C})}$ must possess an abelian normal subgroup whose index is bounded by a function of n alone. In previous papers, the author obtained optimal bounds; in particular, a generic bound <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> <m:mo>!</m:mo> </m:mrow> </m:math> ${(n+1)!}$ was obtained when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>71</m:mn> </m:mrow> </m:math> ${n\ge 71}$ , achieved by the symmetric group S n +1 . In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo form="prefix">Sp</m:mo> <m:mo>(</m:mo> <m:mn>2</m:mn> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>ℂ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> ${\operatorname{Sp}(2n,\mathbb {C})}$ the optimal bound is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mn>60</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msup> <m:mo>·</m:mo> <m:mi>n</m:mi> <m:mo>!</m:mo> </m:mrow> </m:math> ${(60)^{n}\cdot n!}$ , achieved by the wreath product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo form="prefix">SL</m:mo> <m:mn>2</m:mn> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mn>5</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mo form="prefix">wr</m:mo> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> ${\operatorname{SL}_{2}(5)\operatorname{wr}S_{n}}$ acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">O</m:mi> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>ℂ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> ${\mathrm {O}(n,\mathbb {C})}$ , the generic linear group bound of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> <m:mo>!</m:mo> </m:mrow> </m:math> ${(n+1)!}$ is achieved as soon as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>25</m:mn> </m:mrow> </m:math> ${n\ge 25}$ .
This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional …
This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.
The aim of this chapter is to study the subgroup structure of the finite classical groups. The classical groups are defined as certain subgroups of the isometry groups of the …
The aim of this chapter is to study the subgroup structure of the finite classical groups. The classical groups are defined as certain subgroups of the isometry groups of the zero form or of non-degenerate bilinear, sesquilinear or quadratic forms f on finite-dimensional vector spaces over finite fields. According to the classification of such forms and their isometry groups (which can be found for example in [2, §21]), there exist the following nondegenerate possibilities: f is symplectic on an even-dimensional vector space, f is quadratic of maximal Witt index on an odd-dimensional vector space, f is quadratic on a 2n-dimensional vector space, of Witt index n or n - 1, or f is unitary on a vector space over a field with a subfield of index 2. Here, the Witt index of a quadratic form is the maximal dimension of a totally singular subspace.
(1991). On maximal subgroups of finite groups. Communications in Algebra: Vol. 19, No. 8, pp. 2373-2394.
(1991). On maximal subgroups of finite groups. Communications in Algebra: Vol. 19, No. 8, pp. 2373-2394.
The subgroup structure of the finite classical groups has long been the subject of intensive investigation.We explain some of the current issues relating to the study of the maximal subgroups …
The subgroup structure of the finite classical groups has long been the subject of intensive investigation.We explain some of the current issues relating to the study of the maximal subgroups of classical groups.
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the …
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.
We survey some recent results on maximal subgroups of finite classical groups. WE MUST LIVE IN HOPE!!! (R.H. Dye, Newcastle Upon Tyne, Summer 1996)
We survey some recent results on maximal subgroups of finite classical groups. WE MUST LIVE IN HOPE!!! (R.H. Dye, Newcastle Upon Tyne, Summer 1996)
We study irreducible restrictions from modules over alternating groups to proper subgroups, and prove reduction results which substantially restrict the classes of subgroups and modules for which this is possible. …
We study irreducible restrictions from modules over alternating groups to proper subgroups, and prove reduction results which substantially restrict the classes of subgroups and modules for which this is possible. This problem had been solved when the characteristic of the ground field is greater than<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>, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.
We survey some recent results on maximal subgroups of finite classical groups. WE MUST LIVE IN HOPE!!! (R.H. Dye, Newcastle Upon Tyne, Summer 1996)
We survey some recent results on maximal subgroups of finite classical groups. WE MUST LIVE IN HOPE!!! (R.H. Dye, Newcastle Upon Tyne, Summer 1996)
For a finite group G $G$ , let σ ( G ) $\sigma (G)$ be the number of subgroups of G $G$ and σ ι ( G ) $\sigma _\iota …
For a finite group G $G$ , let σ ( G ) $\sigma (G)$ be the number of subgroups of G $G$ and σ ι ( G ) $\sigma _\iota (G)$ the number of isomorphism types of subgroups of G $G$ . Let L = L r ( p e ) $L=L_r(p^e)$ denote a simple group of Lie type, rank r $r$ , over a field of order p e $p^e$ and characteristic p $p$ . If r ≠ 1 $r\ne 1$ , L ≇ 2 B 2 ( 2 1 + 2 m ) $L\not\cong {}^2 B_2(2^{1+2m})$ , there are constants c , d $c,d$ , dependent on the Lie type, such that as r e $re$ grows p ( c − o ( 1 ) ) r 4 e 2 ⩽ σ ι ( L r ( p e ) ) ⩽ σ ( L r ( p e ) ) ⩽ p ( d + o ( 1 ) ) r 4 e 2 . \begin{align*}\hskip6.5pc p^{(c-o(1))r^4e^2} & \leqslant \sigma _{\iota }(L_r(p^e)) \leqslant \sigma (L_r(p^e)) \leqslant p^{(d+o(1))r^4e^2}.\hskip-6.5pc \end{align*} For type A $A$ , c = d = 1 / 64 $c=d=1/64$ . For other classical groups 1 / 64 ⩽ c ⩽ d ⩽ 1 / 4 $1/64\leqslant c\leqslant d\leqslant 1/4$ . For exceptional and twisted groups, 1 / 2 100 ⩽ c ⩽ d ⩽ 1 / 4 $1/2^{100}\leqslant c\leqslant d\leqslant 1/4$ . Furthermore, 2 ( 1 / 36 − o ( 1 ) ) k 2 ) ⩽ σ ι ( Alt k ) ⩽ σ ( Alt k ) ⩽ 24 ( 1 / 6 + o ( 1 ) ) k 2 . \begin{align*}\hskip6.5pc 2^{(1/36-o(1))k^2)} & \leqslant \sigma _{\iota }(\operatorname{Alt }_k) \leqslant \sigma (\operatorname{Alt }_k)\leqslant 24^{(1/6+o(1))k^2}.\hskip-6.5pc \end{align*} For abelian and sporadic simple groups G $G$ , σ ι ( G ) , σ ( G ) ∈ O ( 1 ) $\sigma _{\iota }(G),\sigma (G)\in O(1)$ . In general, these bounds are best possible among groups of the same orders. Thus, with the exception of finite simple groups with bounded ranks and field degrees, the subgroups of finite simple groups are as diverse as possible.
Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥ 0, let H be a proper closed subgroup of G and let V …
Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥ 0, let H be a proper closed subgroup of G and let V be a nontrivial finite dimensional irreducible rational KG-module. We say that (G,H, V) is an irreducible triple if V is irreducible as a KH-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.
We characterize Beauville surfaces of unmixed type with group either PSL(2,p^e) or PGL(2,p^e), thus extending previous results of Bauer, Catanese and Grunewald, Fuertes and Jones, and Penegini and the author.
We characterize Beauville surfaces of unmixed type with group either PSL(2,p^e) or PGL(2,p^e), thus extending previous results of Bauer, Catanese and Grunewald, Fuertes and Jones, and Penegini and the author.
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.
An involution in a finite n-dimensional classical group G over a field of odd order q is called (α, β)-balanced if the dimension of its fixed point subspace is between …
An involution in a finite n-dimensional classical group G over a field of odd order q is called (α, β)-balanced if the dimension of its fixed point subspace is between αn and βn. Balanced involutions play an important role in recent constructive recognition algorithms for finite classical groups in odd characteristic. For a given sequence of conjugacy classes of balanced involutions in G, a c-tuple (g1, . . . , gc) is a class-random sequence from 𝒳 if, for each i = 1, . . . , c, gi is a uniformly distributed random element of , and the gi are mutually independent. We show that there is a number c = c(α, β) such that for large enough n, for a given such sequence 𝒳 of length c, a class-random sequence from 𝒳 generates a subgroup containing the generalized Fitting subgroup of G with probability at least 1 – q–n.
Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in …
Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The minimal dimension of $G$, denoted ${\rm Mindim}(G)$, is the minimal size of a maximal irredundant set of maximal subgroups of $G$. This invariant was recently introduced by Garonzi and Lucchini and they computed the minimal dimension of the alternating groups. In this paper, we prove that ${\rm Mindim}(G) \leqslant 3$ for all finite simple groups, which is best possible, and we compute the exact value for all non-classical simple groups. We also introduce and study two closely related invariants denoted by $α(G)$ and $β(G)$. Here $α(G)$ (respectively $β(G)$) is the minimal size of a set of maximal subgroups (respectively, conjugate maximal subgroups) of $G$ whose intersection coincides with the Frattini subgroup of $G.$ Evidently, ${\rm Mindim}(G) \leqslant α(G) \leqslant β(G)$. For a simple group $G$ we show that $β(G) \leqslant 4$ and $β(G) - α(G) \leqslant 1$, and both upper bounds are best possible.
Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct …
Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.
A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a …
A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is elementary. We then deduce from this result a classification of the thick generalised quadrangles $\mathcal{Q}$ that admit a point-primitive, line-transitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that $\mathcal{Q}$ is flag-transitive and isomorphic to $T_2^*(\mathcal{H})$, where $\mathcal{H}$ is either the regular hyperoval of $\PG(2,4)$ or the Lunelli--Sce hyperoval of $\PG(2,16)$.
Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this …
Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorisations of simple groups (both finite and algebraic) is discussed.
Let V be a d-dimensional vector space over a field of prime order p. We classify the affine transformations of V of order at least p^d/4, and apply this classification …
Let V be a d-dimensional vector space over a field of prime order p. We classify the affine transformations of V of order at least p^d/4, and apply this classification to determine the finite primitive permutation groups of affine type, and of degree n, that contain a permutation of order at least n/4. Using this result we obtain a classification of finite primitive permutation groups of affine type containing a permutation with at most four cycles.
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is …
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful. Connections are made with symmetric function theory, Markov chains, potential theory, Rogers-Ramanujan type identities, quivers, and various measures on partitions.
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.
We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study …
We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).
We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups 3D4(q) in characteristic coprime to q to any proper subgroup is reducible.
We prove that the restriction of any absolutely irreducible representation of Steinberg's triality groups 3D4(q) in characteristic coprime to q to any proper subgroup is reducible.
In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ for the three types of pair $(\Phi,\Delta)$ (the root …
In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ for the three types of pair $(\Phi,\Delta)$ (the root system and its subsystem) such that the group $G(\Delta,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $\sigma$ of the ring $R$ such that $E(\Phi,\Delta,R,\sigma)\le H\le N_{G(\Phi,R)}(E(\Phi,\Delta,R,\sigma))$.
In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$ , then $G$ has a semi-invariant of degree at most $4n^{2}$ …
In 1981, Thompson proved that, if $n\geqslant 1$ is any integer and $G$ is any finite subgroup of $\text{GL}_{n}(\mathbb{C})$ , then $G$ has a semi-invariant of degree at most $4n^{2}$ . He conjectured that, in fact, there is a universal constant $C$ such that for any $n\in \mathbb{N}$ and any finite subgroup $G<\text{GL}_{n}(\mathbb{C})$ , $G$ has a semi-invariant of degree at most $Cn$ . This conjecture would imply that the ${\it\alpha}$ -invariant ${\it\alpha}_{G}(\mathbb{P}^{n-1})$ , as introduced by Tian in 1987, is at most $C$ . We prove Thompson’s conjecture in this paper.
Let G = G(q) be a Chevalley group defined over a field F q of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut( G …
Let G = G(q) be a Chevalley group defined over a field F q of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut( G ) and the centralizers of these involutions. This study was begun in the context of a different problem.