Thompson's first normal p-complement theorem

History
This theorem was proved by Thompson as part of his Ph.D. thesis at the University of Chicago, and was used as a tool to prove the Frobenius conjecture.

Definition
Let $$p$$ be an odd prime number, and $$G$$ a finite group, with $$p$$-Sylow subgroup $$P$$. Suppose $$A$$ is a subgroup of the automorphism group of $$G$$, such that $$P$$ is $$A$$-invariant. Then suppose the following holds:

"For every $$A$$-invariant normal subgroup $$Q$$ of $$P$$, the elements of order relatively prime to $$p$$ which normalize $$Q$$, also centralize $$Q$$"

Then $$P$$ is a retract, i.e. $$G$$ possesses a normal p-complement.

Related results
The result does not hold for $$p=2$$.