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.

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

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

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