Ulrich Meierfrankenfeld

Quadratic GF 2-modules for sporadic simple groups and alternating groups

Let G he a finite group and V a faithful, finite dimensional GF 2 G-module. We call E ≤G quadratic,if[V,E,E]=1 and |E|≥4, and we say V is quadratic, if G is generated by its quadratic subgroups. It is an easy exercise, but important for what follows, to show that E has to be elementary abelian. Elementary abelian groups acting quadratically come up quite naturally in weak-closure arguments, pushing-up problems and in problems dealing with parabolic systems. As a successor of [MS] we treat the case that G is a sporadic simple group or an alternating group.

NON-FINITARY LOCALLY FINITE SIMPLE GROUPS

In this paper we prove and discuss consequences of the following theorem. Let G be a locally finite simple group. Then one of the following holds: (a) G is finitary. (b) G is of alternating type. (c) There exists a prime p and a Kegel cover {(H i ,M i ) | i ∊ I} such that G is of p-type and, for all i in I, H i /M i is a projective special linear group.

Fixed point spaces, primitive character degrees and conjugacy class sizes

Let G be a finite group that acts on a nonzero finite dimensional vector space V over an arbitrary field. Assume that V is completely reducible as a G-module, and that G fixes no nonzero vector of V. We show that some element g ∈ G has a small fixed-point space in V. Specifically, we prove that we can choose g so that dim C V (g) < (1/p)dim V, where p is the smallest prime divisor of |G|.

F-stability in finite groups

Let G be a finite group, S ∈ Syl P (G), and S be the set subgroups containing S. For M ∈ S and V = Ω 1 Z(Op(M)), the paper discusses the action of M on V. Apart from other results, it is shown that for groups of parabolic characteristic p either S is contained in a unique maximal p-local subgroup, or there exists a maximal p-local subgroup in M ∈ S such that V is a nearly quadratic 2F-module for M.

Local identification of spherical buildings and finite simple groups of lie type

We give a short proof of the uniqueness of finite spherical buildings of rank at least 3 in terms of the structure of the rank 2 residues and use this result to prove a result making it possible to identify an arbitrary finite group of Lie type from knowledge of its “parabolic structure” alone. Our proof also involves a connection between loops, “Latin chamber systems” and buildings.

Repeated and final commutators in group actions

Let G be a finite group and suppose that A acts via automorphisms on G. The repeated commutators are the subgroups [G,A,A, . . . , A], where there is some positive number of commutations by A, and the final commutator is the smallest of these repeated commutators. We show that if [G,A] is nilpotent, then the final commutator is normal in G. Also, in general, if K is an arbitrary repeated commutator and P is the permutation group induced by the action of A on the left cosets of K in G, we relate the structure of P to the structure of [G,A].

Classification of Finite Completions of a Class of Locally D 8 Amalgams, II

Let 𝒜 = (A 1, A 12, A 2) be a locally D 8 amalgam with a finite completion G. Suppose that A 1 ∈ Syl 2(G). We show that under these conditions |A 1| ≤25, or N A 1 (A 12) is Abelian. As applications of our results, we determine all the finite completions G, up to O(G), in the case where N A 1 (A 12) is non-Abelian.

On normalizers of nilpotent subgroups

The set of all nilpotent subgroups of a group is an example of a system satisfying (C), (I) and (P). Examples of NSS’s are the set of p-subgroups of a finite group (p a prime), the set of closed unipotent subgroups of an algebraic group, and the set of maximal cyclic subgroups plus the trivial group in a free group. To state our main theorem we introduce a good portion of the notations used in this paper. Let Σ be a set of subgroups of G. Σ∗ is the set of maximal elements of Σ (with respect to inclusion). The elements of Γ∗ are called maximal Γ-subgroups. Σ∗ is the set of minimal non-trivial elements of Σ. The elements of Γ∗ are called minimal Γ-subgroups.

Extensions of Periodic Linear Groups with Finite Unipotent Radical

Abstract A group is called p-linear if it is isomorphic to a subgroup of GL(n,K) for some field K of characteristic p and some integer n. Let H be a normal subgroup of G and assume that both H and G/H are periodic and p-linear. In addition, assume that both H and G/H have finite unipotent radicals and that the Hirsch-Plotkin radical of H is C˘ernikov. The main result of this article is a proof that under these assumptions G is p-linear. An example is provided showing the result is false if the assumption regarding the Hirsch-Plotkin radical is removed.