Thompson's first normal p-complement theorem

This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number $p$.
View other normal p-complement theorems

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 $p=2$. Further information: Thompson's normal p-complement theorem fails at the prime two

References

• Normal p-complements for finite groups by John G. Thompson, Math. Zeitschr. 72, 332--354 (1960)
• Finite groups with fixed-point-free automorphisms of prime order by John G. Thompson, Proc. Nat. Acad. Sci. 45, 578-581 (1959)