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 SL("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 .