P-stable group

Template:Short description Template:Distinguish Template:Lowercase In finite group theory, a p-stable group for an odd prime p is a finite group satisfying a technical condition introduced by Template:Harvs 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 Template:Harv.

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 $ℳ_{p} (G)$ as the set of all p-subgroups of G maximal with respect to the property that $O_{p} (M) = 1$ .

2. Let G be a finite group and p an odd prime. Then G is called p-stable if every element of $ℳ_{p} (G)$ 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 $g C_{H} (P) \in O_{p} (H / C_{H} (P))$ .

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 $C_{G} (P) ⩽ P$ , then $Z (J_{0} (S))$ is a characteristic subgroup of G, where $J_{0} (S)$ is the subgroup introduced by John Thompson in Template:Harv.

References

P-stable group

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P-stable group

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References

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