Exponent of a finite group with a four-group of automorphisms

Abstract

Let A be a group isomorphic with either S4, the symmetric group on four symbols, or D8, the dihedral group of order 8. Let V be a normal four-subgroup of A and α an involution in \({A\setminus V}\). Suppose that A acts on a finite group G in such a manner that CG(V) = 1 and CG(α) has exponent e. We show that if \({A\cong S_4}\) then the exponent of G is e-bounded and if \({A\cong D_8}\) then the exponent of the derived group G′ is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.