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 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.

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 $$O_{p'\!}(G)$$ 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.