Michael Aschbacher

On the maximal subgroups of the finite classical groups.

(1.1) Definition Let 1 6= G be a group. A subgroup M of G is said to be maximal if M 6= G and there exists no subgroup H such that M < H < G. IfG is finite, by order reasons every subgroupH 6= G is contained in a maximal subgroup. If M is maximal in G, then also every conjugate gMg−1 of M in G is maximal. Indeed gMg−1 < K < G =⇒ M < g−1Kg < G. For this reason the maximal subgroups are studied up to conjugation. (1.2) Lemma Let G = G′ and let M be a maximal subgroup of G. Then:

A Characterization of Chevalley Groups Over Fields of Odd Order

THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian Sylow 2-subgroups and z is the unique involution in K. Assume for each 2-element k C K that kG n C(z) C N(K) and for each g e C(z) N(K), that [K, Kg]<! O(C(z)). Then F*(G) is a Chevalley group of odd characteristic, M11, M12 or SP6(2). COROLLARY II. Let G be a finite group with F*(G) simple and let K be tightly embedded in G such that K has quaternion Sylow 2-subgroups. Then F*(G) is a Chevalley group of odd characteristic, M11, or M12. COROLLARY III. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a 2-component or solvable 2-component of CQ(z) of 2-rank 1, containing z. Then F*(G) is a Chevalley group of odd characteristic or M,1. Theorem I follows from Theorems 1 through 8, stated in Section 2, which supply more specific information under more general hypotheses. Corollary II follows directly from Theorem I. Corollary III follows from Theorem I and Theorem 3 in [3]. All Chevalley groups with the exception of L2(q) and 2G2(q) satisfy the hypotheses of Theorem I. Terminology and notation are defined in Section 2. The possibility of such a theorem was first suggested by J.G. Thompson in January 1974, during his lectures at the winter meeting of the American Mathematical Society in San Francisco. At the same time Thompson also pointed out the significance of a certain section of the group, which is crucial to the proof. We have taken the liberty of referring to this section as the Thompson group of G; see Section 2 for its definition. The theorem finds its motivation in the study of component-type groups. Some applications to this theory are described in [6]. The remainder of this

On collineation groups of symmetric block designs

Abstract The question of possible collineation groups for a symmetric design with given parameters ( v , k , λ ) is considered. A new design with parameters (79, 13, 2) is constructed.

On finite groups generated by odd transpositions. I

Publisher Summary This chapter discusses the finite groups generated by odd transpositions. Let ω be a set of integers and G a finite group. As defined by B. Fischer, a subset D of involutions of G is a set of ω-transpositions if G = , DG = D, and |uv| ∈ {1, 2} ω for all u, v ∈ D. D is a set of odd transpositions if ω is the set of all odd integers. Let D be a set of odd transpositions of G and u, v ɛ D. Then either [u, v] = 1 or u is conjugate to v in .

Simple connectivity ofp-group complexes

We investigate the simple connectivity ofp-subgroup complexes of finite groups.

Groups Generated by a Class of Elements of Order 3

Abstract : The Conway group which is the group of automorphisms of the 24 dimensional Leech lattice is generated by a class of elements of order 3 with the property that any two of them either commute or generate SL2(3), SL2(5) or the alternating groups A4, A5 which are isomorphic to SL2(3) and SL2(5) modulo a center of order 2. Such a class of order 3 is also a special case of John Thompsons Quadratic pairs for the prime 3. The paper is restricted to elements of order 3 in which any two either commute, generate A4 or SL2(3). It is possible to describe these groups completely. (Author)

Some Multilinear Forms with Large Isometry Groups

Many groups are best described as the group of automorphisms of some natural object. I’m interested in obtaining such descriptions of the finite simple groups, and more generally descriptions of the groups of Lie type over arbitrary fields. The representation of the alternating group of degree n as the group of automorphisms of a set of order n is an excellent example of such a description. The representation of the classical groups as the isometry groups of bilinear or sequilinear forms is another.

On abelian quotients of primitive groups

It is shown that if G is a primitive permutation group on a set of size n , then any abelian quotient of G has order at most n . This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if MjK is a minimal extension and L/K is an abelian extension contained in the normal closure of M, then the degree of L/K is at most the degree of M/K.

The limitations of nice mutually unbiased bases

Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets.

On finite groups generated by odd transpositions, III

Publisher Summary This chapter discusses the finite groups generated by odd transpositions. Let ω be a set of integers and G a finite group. As defined by B. Fischer, a subset D of involutions of G is a set of ω-transpositions if G = , D G = D, and |uv| ∈ {1, 2} ω for all u, v ∈ D. D is a set of odd transpositions if ω is the set of all odd integers. Let D be a set of odd transpositions of G and u, v ɛ D. Then either [u, v] = 1 or u is conjugate to v in .