Finite Groups have local Non-Schur Centralizers.
In this paper it is proved that a finite group G with an automorphism of prime order r, such that is contained in a nilpotent subgroup H, with , is nilpotent provided that either is odd or, if is even, then r is not a Fermât prime.
We describe finite groups which contain just one conjugate class of self-normalizing subgroups.