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 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.

Not to be confused with Stable group.

Definitions

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 ${\displaystyle O_{p}(G)}$. Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that ${\displaystyle O_{p^{\prime }}(G)}$ is a normal subgroup of G. Suppose that ${\displaystyle x\in N_{G}(P)}$ and ${\displaystyle {\bar {x}}}$ is the coset of ${\displaystyle C_{G}(P)}$ containing x. If ${\displaystyle [P,x,x]=1}$, then ${\displaystyle {\overline {x}}\in O_{n}(N_{G}(P)/C_{G}(P))}$.

Now, define ${\displaystyle {\mathcal {M}}_{p}(G)}$ as the set of all p-subgroups of G maximal with respect to the property that ${\displaystyle O_{p}(M)\not =1}$.

2. Let G be a finite group and p an odd prime. Then G is called p-stable if every element of ${\displaystyle {\mathcal {M}}_{p}(G)}$ 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 ${\displaystyle F^{*}(H)=O_{p}(H)}$ and, whenever P is a normal p-subgroup of H and ${\displaystyle g\in H}$ with ${\displaystyle [P,g,g]=1}$, then ${\displaystyle gC_{H}(P)\in O_{p}(H/C_{H}(P))}$.

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 ${\displaystyle C_{G}(P)\leqslant P}$, then ${\displaystyle Z(J_{0}(S))}$ is a characteristic subgroup of G, where ${\displaystyle J_{0}(S)}$ is the subgroup introduced by John Thompson in (Thompson 1969, pp. 149–151).

See also

• p-stability is used as one of the conditions in Glauberman's ZJ theorem.
• Quadratic pair
• p-constrained group
• p-solvable group

References

• Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807
• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13 (2): 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683
• Gorenstein, D.; Walter, John H. (1964), "On the maximal subgroups of finite simple groups", Journal of Algebra, 1 (2): 168–213, doi:10.1016/0021-8693(64)90032-8, ISSN 0021-8693, MR 0172917
• Gorenstein, D.; Walter, John H. (1965), "The characterization of finite groups with dihedral Sylow 2-subgroups. I", Journal of Algebra, 2: 85–151, doi:10.1016/0021-8693(65)90027-X, ISSN 0021-8693, MR 0177032
• Gorenstein, D.; Walter, John H. (1965), "The characterization of finite groups with dihedral Sylow 2-subgroups. II", Journal of Algebra, 2 (2): 218–270, doi:10.1016/0021-8693(65)90019-0, ISSN 0021-8693, MR 0177032
• Gorenstein, D.; Walter, John H. (1965), "The characterization of finite groups with dihedral Sylow 2-subgroups. III", Journal of Algebra, 2 (3): 354–393, doi:10.1016/0021-8693(65)90015-3, ISSN 0021-8693, MR 0190220
• 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, doi:10.1090/S0273-0979-1979-14551-8, ISSN 0002-9904, MR 0513750
• Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, MR 0569209