George Glauberman

Central Elements in Core-Free Groups

Let G be a finite group. By using transfer methods, several authors have determined sufficient conditions for G to have a nontrivial Abelian factor group [9, Chapter 141. In this paper we establish some sufficient conditions for G to have a nontrivial center or a nontrivial normal subgroup of odd order. For every finite group G, let Z(G) be the center of G. Following Brauer [I, 21, we define the COY~ of G, K(G), to be the largest normal subgroup of odd order in G, and we say that G is core-free if K(G) = 1. Define Z*(G) to be the subgroup of G containing K(G) for which Z*(G)/K(G) = Z(G/K(G)). Note that Z*(G) = Z(G) if G is core-free. For every pair of subgroups J and K of G, denote the centralizer and normalizer of J in K by CK(J) and NK(J), respectively. Similarly, denote by C,(x) the centralizer in K of an element x of G. The main result of this paper is the following proposition:

Prime-Power Factor Groups of Finite Groups. II

Theorem A. Suppose S is a p-subgroup of G, T ~_ S and T < G. Assume that Ko~(S)~= K~(T ) or K~(S)=[= K~(T) (e.g., if K~(S):~G or K~(S):~G). Then there exists g ~ S T satisfying the following conditions: (a) For every chief factor X / Y of G such that X ~_ T, we have [X, g; 4] ~ Y (b) I f T= Op(G) and C(T)~_ T,, then there exists a chieffactor X / Y of G such that X ~_ T and IX, g, g] ~_ Y and [X, g] ~ Y

Local analysis for the odd order theorem

Part I. Preliminary Results: 1. Notation and elementary properties of solvable groups 2. General results on representations 3. Actions of Frobenius groups and related results 4. p-Groups of small rank 5. Narrow p-groups 6. Additional results Part II. The Uniqueness Theorem: 7. The transitivity theorem 8. The fitting subgroup of a maximal subgroup 9. The uniqueness theorem Part III. Maximal Subgroups: 10. The subgroups Ma and Me 11. Exceptional maximal subgroups 12. The subgroup E 13. Prime action Part IV. The Family of All Maximal Subgroups of G: 14. Maximal subgroups of type p and counting arguments 15. The subgroup Mf 16. The main results Appendix Prerequisites and p-stability.

Quadratic Elements in Unipotent Linear Groups

Let G be a group, A be a commutative ring with unity, and M be a unitary (right) AG-module that possesses a composition series with respect to A. Assume that G acts faithfully on M. Let Q be the set of all nonidentity elements g of G such that M(g - 1)2 = 0; such elements will be called quadratic elements of G. For each g E Q, let d(g) be the composition length of M(g - 1) with respect to A. If Q is not empty, let d = min d(g), gEQ Qd = {gEQ I d(g) = d}. Recently [2], John Thompson has proved that G is a known group if G acts irreducibly on M, Q generates G, and A = Zp for some prime p greater than three. In the course of his proof, he shows that if two elements e and f of Qd generate a p-group, then this p-group has nil potence class at most two (Section 16). In this paper, we obtain an alternate proof of this result by examining only the group generated by e and f and using the fact that it is unipotent. We also obtain some other intermediate results of [2]. Our results are valid if A = Z3 or, in fact, if A is any field of characteristic other than two. Since no change in the proof is required, we have adopted a more general hypothesis.

Characters of finite groups with dihedral Sylow 2-subgroups☆

In this article, we give alternate proofs of some intermediate results in the Gorenstein-Walter classification [6] of groups with dihedral Sylow 2subgroups. These intermediate results will be applied in an alternate proof [ 1 ] of the entire classification by the first author. Our proofs rely only on elementary character theory and do not use block theory, but they are motivated by the related work of Brauer [2] and of Gorenstein and Walter [6, Propositions l-41, which was obtained by block theory. Now we describe the situation which will be assumed in the following sections and in the three theorems stated below. For elementary properties of groups with dihedral Sylow 2-subgroups the reader is referred to [S, Section 7.71.

Control of fixed points and existence and uniqueness of centric linking systems

A. Chermak has recently proved that to each saturated fusion system over a finite p-group, there is a unique associated centric linking system. B. Oliver extended Chermak’s proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino–Priddy conjecture concerning the p-completed classifying spaces of finite groups. Our main tool is a 1971 result of the first author on control of fixed points by p-local subgroups. This result is directly applicable for odd primes, and we show how a slight variation of it allows applications for

A partial extension of Lazard's correspondence for finite p-groups

p=2

Nilpotence of finite Moufang 2-loops

p=2 in the presence of offenders.