In group theory, a branch of mathematics, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup.
This implies that if p is the smallest prime dividing the order of a group G and the Sylow p-subgroup is cyclic, then G has a normal p-complement.
For odd primes p Thompson found such a strengthened criterion: in fact he did not need all characteristic subgroups, but only two special ones.
The result fails for p = 2 as the simple group PSL2(F7) of order 168 is a counterexample.
Thompson's normal p-complement theorem used conditions on two particular characteristic subgroups of a Sylow p-subgroup.