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p-stable group

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An odd prime p is a finite group Not to be confused with 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 ) {\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 O p ( G ) {\displaystyle O_{p'\!}(G)} is a normal subgroup of G. Suppose that x N G ( P ) {\displaystyle x\in N_{G}(P)} and x ¯ {\displaystyle {\bar {x}}} is the coset of C G ( P ) {\displaystyle C_{G}(P)} containing x. If [ P , x , x ] = 1 {\displaystyle =1} , then x ¯ O n ( N G ( P ) / C G ( P ) ) {\displaystyle {\overline {x}}\in O_{n}(N_{G}(P)/C_{G}(P))} .

Now, define M p ( G ) {\displaystyle {\mathcal {M}}_{p}(G)} as the set of all p-subgroups of G maximal with respect to the property that O p ( M ) 1 {\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 M p ( G ) {\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 F ( H ) = O p ( H ) {\displaystyle F^{*}(H)=O_{p}(H)} and, whenever P is a normal p-subgroup of H and g H {\displaystyle g\in H} with [ P , g , g ] = 1 {\displaystyle =1} , then g C H ( P ) O p ( H / C H ( P ) ) {\displaystyle 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 ) P {\displaystyle C_{G}(P)\leqslant P} , then Z ( J 0 ( S ) ) {\displaystyle Z(J_{0}(S))} is a characteristic subgroup of G, where J 0 ( S ) {\displaystyle J_{0}(S)} is the subgroup introduced by John Thompson in (Thompson 1969, pp. 149–151).

See also

References

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