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).
- p-stability is used as one of the conditions in Glauberman's ZJ theorem.
- Quadratic pair
- p-constrained group
- p-solvable group
- Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, ISSN 0008-414X, MR 0230807, doi:10.4153/cjm-1968-107-2
- Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13: 149–151, ISSN 0021-8693, MR 0245683, doi:10.1016/0021-8693(69)90068-4
- Gorenstein, D.; Walter, John H. (1964), "On the maximal subgroups of finite simple groups", Journal of Algebra, 1: 168–213, ISSN 0021-8693, MR 0172917, doi:10.1016/0021-8693(64)90032-8
- Gorenstein, D.; Walter, John H. (1965), "The characterization of finite groups with dihedral Sylow 2-subgroups. I", Journal of Algebra, 2: 85–151, ISSN 0021-8693, MR 0177032, doi:10.1016/0021-8693(65)90027-X
- Gorenstein, D.; Walter, John H. (1965), "The characterization of finite groups with dihedral Sylow 2-subgroups. II", Journal of Algebra, 2 (2): 218–270, ISSN 0021-8693, MR 0177032, doi:10.1016/0021-8693(65)90019-0
- Gorenstein, D.; Walter, John H. (1965), "The characterization of finite groups with dihedral Sylow 2-subgroups. III", Journal of Algebra, 2 (3): 354–393, ISSN 0021-8693, MR 0190220, doi:10.1016/0021-8693(65)90015-3
- Gorenstein, D. (1979), "The classification of finite simple groups. I. Simple groups and local analysis", American Mathematical Society. Bulletin. New Series, 1 (1): 43–199, ISSN 0002-9904, MR 513750, doi:10.1090/S0273-0979-1979-14551-8
- Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 569209