p-stable group

In finite group theory, a p-stable group for an odd prime p is a finite group satisfying a technical condition introduced by Gorenstein and Walter (1964, p.169, 1965) in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups.

Definitions

There are several equivalent definitions of a p-stable group.

First definition.

We give definition of a p-stable group in two parts. The definition used here comes from (Glauberman 1968, p. 1104).

1. Let p be an odd prime and G be a finite group with a nontrivial p-core $O_{p}(G)$ . Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that $O_{p^{\prime }}(G)$ is a normal subgroup of G. Suppose that $x\in N_{G}(P)$ and ${\bar {x}}$ is the coset of $C_{G}(P)$ containing x. If $[P,x,x]=1$ , then ${\overline {x}}\in O_{n}(N_{G}(P)/C_{G}(P))$ .

Now, define ${\mathcal {M}}_{p}(G)$ as the set of all p-subgroups of G maximal with respect to the property that $O_{p}(M)\not =1$ .

2. Let G be a finite group and p an odd prime. Then G is called p-stable if every element of ${\mathcal {M}}_{p}(G)$ is p-stable by definition 1.

Second definition.

Let p be an odd prime and H a finite group. Then H is p-stable if $F^{*}(H)=O_{p}(H)$ and, whenever P is a normal p-subgroup of H and $g\in H$ with $[P,g,g]=1$ , then $gC_{H}(P)\in O_{p}(H/C_{H}(P))$ .

Properties

If p is an odd prime and G is a finite group such that SL₂(p) is not involved in G, then G is p-stable. If furthermore G contains a normal p-subgroup P such that $C_{G}(P)\leqslant P$ , then $Z(J_{0}(S))$ is a characteristic subgroup of G, where $J_{0}(S)$ is the subgroup introduced by John Thompson in (Thompson 1969, pp. 149–151).