p-stable group

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 ${\displaystyle O_{p}(G)}$. Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that ${\displaystyle O_{p'\!}(G)}$ 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 ${\displaystyle {\overline {x}}\in O_{n}(N_{G}(P)/C_{G}(P))}$.

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

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

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 ${\displaystyle C_{G}(P)\leqslant P}$, then ${\displaystyle Z(J_{0}(S))}$ is a characteristic subgroup of G, where ${\displaystyle J_{0}(S)}$ is the subgroup introduced by John Thompson in (Thompson 1969, pp. 149–151).

• p-stability is used as one of the conditions in Glauberman's ZJ theorem.
• Quadratic pair
• p-constrained group
• p-solvable group

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• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13 (2): 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683
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• Gorenstein, D. (1979), "The classification of finite simple groups. I. Simple groups and local analysis", Bulletin of the American Mathematical Society, New Series, 1 (1): 43–199, doi:10.1090/S0273-0979-1979-14551-8, ISSN 0002-9904, MR 0513750
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