Thompson's second 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

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with Thompson's first normal p-complement theorem

Statement

Suppose $p$ is an odd prime number and $G$ is a Strongly p-solvable group (?) that is also p-core-free. Suppose $P$ is a $p$-Sylow subgroup of $G$. Then:

$G=C_{G}(Z(P))N_{G}(J^{*}(P))$

where $Z(P)$ is the center of $P$, $C_{G}$ denotes the centralizer, $J^{*}(P)$ is the join of abelian subgroups of maximum rank, and $N_{G}$ denotes the normalizer.

Related facts

• Glauberman-Thompson normal p-complement theorem