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 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.

Glauberman

We give definition of a p-stable group in two parts. The definition used here comes from.
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.

Alternative

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.

Properties

If p is an odd prime and G is a finite group such that SL₂ 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.