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.
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).
1. Let p be an odd prime and G be a finite group with a nontrivial p-core [math]\displaystyle{ O_p(G) }[/math]. Then G is p-stable if it satisfies the following condition: Let P be an arbitrary p-subgroup of G such that [math]\displaystyle{ O_{p'\!}(G) }[/math] is a normal subgroup of G. Suppose that [math]\displaystyle{ x \in N_G(P) }[/math] and [math]\displaystyle{ \bar x }[/math] is the coset of [math]\displaystyle{ C_G(P) }[/math] containing x. If [math]\displaystyle{ [P,x,x]=1 }[/math], then [math]\displaystyle{ \overline{x}\in O_n(N_G(P)/C_G(P)) }[/math].
Now, define [math]\displaystyle{ \mathcal{M}_p(G) }[/math] as the set of all p-subgroups of G maximal with respect to the property that [math]\displaystyle{ O_p(M)\not= 1 }[/math].
2. Let G be a finite group and p an odd prime. Then G is called p-stable if every element of [math]\displaystyle{ \mathcal{M}_p(G) }[/math] 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 [math]\displaystyle{ F^*(H)=O_p(H) }[/math] and, whenever P is a normal p-subgroup of H and [math]\displaystyle{ g \in H }[/math] with [math]\displaystyle{ [P,g,g]=1 }[/math], then [math]\displaystyle{ gC_H(P)\in O_p(H/C_H(P)) }[/math].
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 [math]\displaystyle{ C_G(P)\leqslant P }[/math], then [math]\displaystyle{ Z(J_0(S)) }[/math] is a characteristic subgroup of G, where [math]\displaystyle{ J_0(S) }[/math] is the subgroup introduced by John Thompson in (Thompson 1969).
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, http://www.cms.math.ca/cjm/v20/p1101
- 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
- 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
- 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
- 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
- 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
- Gorenstein, D. (1979), "The classification of finite simple groups. I. Simple groups and local analysis", Bulletin of the American Mathematical Society, New Series 1 (1): 43–199, doi:10.1090/S0273-0979-1979-14551-8, ISSN 0002-9904
- Gorenstein, D. (1980), Finite groups (2nd ed.), New York: Chelsea Publishing Co., ISBN 978-0-8284-0301-6, https://www.ams.org/bookstore-getitem/item=CHEL-301-H
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