# Thompson's second normal p-complement theorem

From Groupprops

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 .

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

where is the center of , denotes the centralizer, is the join of abelian subgroups of maximum rank, and denotes the normalizer.