p-stable group

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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.

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 . Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that is a normal subgroup of G. Suppose that and is the coset of containing x. If , then .

Now, define as the set of all p-subgroups of G maximal with respect to the property that .

2. Let G be a finite group and p an odd prime. Then G is called p-stable if every element of 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 and, whenever P is a normal p-subgroup of H and with , then .


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 , then is a characteristic subgroup of G, where is the subgroup introduced by John Thompson in (Thompson 1969, pp. 149–151).

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