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.

;First definition.

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 $O_p(G)$. Then ''G'' is ''p''-stable if it satisfies the following condition: Let ''P'' be an arbitrary ''p''-subgroup of ''G'' such that $O_(G)$ is a normal subgroup of ''G''. Suppose that $x \in N_G(P)$ and $\bar x$ is the coset of $C_G(P)$ containing ''x''. If ,x,x1, then $\overline\in O_n(N_G(P)/C_G(P))$.

Now, define $\mathcal_p(G)$ as the set of all ''p''-subgroups of ''G'' maximal with respect to the property that $O_p(M)\not= 1$.

2. Let ''G'' be a finite group and ''p'' an odd prime. Then ''G'' is called ''p''-stable if every element of $\mathcal_p(G)$ 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 $F^*(H)=O_p(H)$ and, whenever ''P'' is a normal ''p''-subgroup of ''H'' and $g \in H$ with ,g,g1, then $gC_H(P)\in O_p(H/C_H(P))$.

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 $C_G(P)\leqslant P$, then $Z(J_0(S))$ is a characteristic subgroup of ''G'', where $J_0(S)$ is the subgroup introduced by John Thompson in .

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

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*{{Citation , last1=Gorenstein , first1=D. , author1-link=Daniel Gorenstein , title=Finite groups , url=https://www.ams.org/bookstore-getitem/item=CHEL-301-H , publisher=Chelsea Publishing Co. , location=New York , edition=2nd , isbn=978-0-8284-0301-6 , mr=569209 , year=1980

Finite groups