Generation of simple groups (original) (raw)

COROLLARY 2. If G is a finite group, then G = (S, T) for some SE Syl,(G) and some solvable subgroup T (containing a Sylow 2 subgroup). A slightly stronger version of Theorem A (Theorem 4.3) gives a bound on the dimension of the first cohomology group over fields of odd characteristic. We complete the argument for even characteristic and obtain: THEOREM B. Let G be a finite group and K a field of charcteristic p. Zf G acts faithfully on the irreducible KG-module V, then dim H'(G, V) < 3 dim V. This improves the bound in [AG2, Theorem A]. An immediate consequence of Theorem B and [Gr2, p. 361 is COROLLARY 3. Let G be a finite simple group and M a minimal relation module for G. Then ZG is a summand of Mt3'. The fact that M(') has a free summand for some t is a consequence of the weaker bound in [AG2]. This has several consequences (see [K, L]). Since d(G) = 2 for G simple, it follows by [Gr2, Proposition 6.21 that M has no projective summand. We conjecture that 3 can be improved to t. This would imply that M (2) has a free summand. There are low dimensional examples where + is achieved. Indeed, it is quite possible that dim H'(G, V) < 1, for some fixed 1 (possibly 1= 2) for all faithful absolutely irreducible modules V. One can obtain a variation of Corollary 2 from the next result. This can be viewed as a partial dual to Cauchy's theorem. THEOREM C. Let G be a finite group of even order. Let O(G) be the maximal normal subgroup of G of odd order, and set G = G/O(G). Then either G has a maximal subgroup of even index or A = O,(G) = Q(G) and G/A g A,. COROLLARY 4. Every finite group can be generated by a 2-subgroup and an odd subgroup.

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On the essential dimension of a finite group

Compositio Mathematica, 1997

Let f (x) = aix i be a monic polynomial of degree n whose coefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) is generating (for the symmetric group) if it can be obtained from f (x) by a non-degenerate Tschirnhaus transformation. We show that the minimal number d k (n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13-th problem. Our approach to this question (and generalizations) is based on the idea of the "essential dimension" of a finite group G: the smallest possible dimension of an algebraic G-variety over k to which one can "compress" a faithful linear representation of G. We show that d k (n) is just the essential dimension of the symmetric group Sn. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

Adequate groups of low degree

Algebra & Number Theory, 2015

The notion of adequate subgroups was introduced by Jack Thorne. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown by Guralnick, Herzig, Taylor, and Thorne that if the dimension is small compared to the characteristic, then all absolutely irreducible representations are adequate. Here we extend that result by showing that, in almost all cases, absolutely irreducible kG-modules in characteristic p whose irreducible G +summands have dimension less than p (where G + denotes the subgroup of G generated by all p-elements of G) are adequate. 1. Introduction 78 2. Linear groups of low degree 81 3. Weak adequacy for SL 2 ‫ކ(‬ p) 85 4. Weak adequacy for Chevalley groups 89 5. Weak adequacy in cross-characteristic 98 6. Weak adequacy for special linear groups 114 7. Extensions and self-extensions, I: Generalities 127 8. Indecomposable representations of SL 2 (q) 131 9. Finite groups with indecomposable modules of small dimension 133 10. Extensions and self-extensions, II 140 References 144

Groups of small homological dimension and the Atiyah Conjecture

2004

A group G has homological dimension less or equal to 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L^2-Betti numbers. We also show that a finitely generated elementary amenable group G of cohomological dimension less or equal to 2 possesses a finite 2-dimensional model for BG and in particular that

On the dimension growth of groups

Journal of Algebra, 2011

We prove that the Thompson group F has exponential dimension growth. We also prove that every solvable finitely generated subgroups of F has polynomial dimension growth while some elementary amenable subgroups of F and some solvable groups of class 3 have dimension growth at least exp( √ n).

Essential dimension of algebraic groups, including bad characteristic

Archiv der Mathematik, 2016

We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group G of rank at least two is at most dim G − 2(rank G) − 1. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra.

Maximal subgroups of exceptional groups and Quillen's dimension

arXiv (Cornell University), 2023

A. Given a finite group and a prime , let A () be the poset of nontrivial elementary abelian-subgroups of. The group satisfies the Quillen dimension property at if A () has non-zero homology in the maximal possible degree, which is the-rank of minus 1. For example, D. Quillen showed that solvable groups with trivial-core satisfy this property, and later, M. Aschbacher and S.D. Smith provided a list of all-extensions of simple groups that may fail this property if is odd. In particular, a group with this property satisfies Quillen's conjecture: has trivial-core and the poset A () is not contractible. In this article, we focus on the prime = 2 and prove that the 2-extensions of the exceptional finite simple groups of Lie type in odd characteristic satisfy the Quillen dimension property, with only finitely many exceptions. We achieve these conclusions by studying maximal subgroups and usually reducing the problem to the same question in small linear groups, where we establish this property via counting arguments. As a corollary, we reduce the list of possible components in a minimal counterexample to Quillen's conjecture at = 2.

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