P-stable group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

Let $G$ be a finite group and $p$ be a prime number. We say that $G$ is a $p$-stable group if either $G$ is trivial or $G$ has a nontrivial normal $p$-subgroup and $G$ satisfies the following:

Suppose $P$ is a $p$-subgroup of $G$ such that $O_{p'}(G)P$ is normal in $G$. Then, if $A$ is a $p$-subgroup of $N_{G}(P)$ with the property that $[[P,A],A]$ is trivial, we have:

$AC_{G}(P)/C_{G}(P)\leq O_{p}(N_{G}(P)/C_{G}(P))$.

Relation with other properties

Stronger properties

• Strongly p-solvable group: For proof of the implication, refer strongly p-solvable implies p-stable and for proof of its strictness (i.e. the reverse implication being false) refer p-stable not implies strongly p-solvable.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 268, Chapter 8 (p-constrained and p-stable groups), Section 8.1 (p-constraint and p-stability), ^{More info}