Thompson's second normal p-complement theorem

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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.

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