P stable group

Definitions

There are several equivalent definitions of a p-stable group.

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 ) . 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 ∈ N G ( P ) and x ¯ is the coset of C G ( P ) containing x. If [ P , x , x ] = 1 , then x ¯ ∈ 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.

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 ∈ H with [ P , g , g ] = 1 , then g C H ( P ) ∈ 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 (Thompson 1969, pp. 149–151).