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.
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 formula_1. Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that formula_2 is a normal subgroup of G. Suppose that formula_3 and formula_4 is the coset of formula_5 containing x. If formula_6, then formula_7.
Now, define formula_8 as the set of all p-subgroups of G maximal with respect to the property that formula_9.
2. Let G be a finite group and p an odd prime. Then G is called p-stable if every element of formula_8 is p-stable by definition 1.
Let p be an odd prime and H a finite group. Then H is p-stable if formula_11 and, whenever P is a normal p-subgroup of H and formula_12 with formula_13, then formula_14.
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 formula_15, then formula_16 is a characteristic subgroup of G, where formula_17 is the subgroup introduced by John Thompson in .