Groups of automorphisms with TNI-centralizers.

A subgroup H of a finite group G is called a TNI -subgroup if N G ( H ) ∩ H g = 1 for any g ∈ G \ N G ( H ) . Let A be a group acting on G by automorphisms where C G ( A ) is a TNI -subgroup of G . We prove that G is solvable if and only if C G ( A ) is solvable, and determine some bounds for the nilpotent length of G in terms of the nilpotent length of C G ( A ) under some additional assumptions. We also study the action of a Frobenius group FH of automorphisms on a group G if the set of fixed points C G ( F ) of the kernel F forms a TNI -subgroup, and obtain a bound for the nilpotent length of G in terms of the nilpotent lengths of C G ( F ) and C G ( H ) . [ABSTRACT FROM AUTHOR]