Corollary of Thompson transitivity theorem

Statement

Let $G$ be a finite group and $p$ is a prime number dividing the order of $G$. Suppose $G$ is a Group in which every p-local subgroup is p-constrained (?). Suppose $P$ is a $p$-Sylow subgroup of $G$ and $A$ is maximal among abelian normal subgroups in $P$, with the rank of $A$ at least three.

Suppose $q$ is a prime number not equal to $p$.

Then:

1. $P$ normalizes at least one maximal $A$-invariant $q$-subgroup of $G$.
2. If $P$ normalizes no nontrivial $p'$-subgroups of $G$, then neither does $A$.

Facts used

1. Thompson transitivity theorem

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 293, Theorem 5.6, Chapter 8, ^{More info}