Coprime automorphisms of finite groups

Let G G be a finite group admitting a coprime automorphism α \alpha of order e e . Denote by I G ( α ) I_G(\alpha ) the set of commutators g − 1 g α g^{-1}g^\alpha , where g ∈ G g\in G , and by [ G , α ] [G,\alpha ] the subgroup generated by I G ( α ) I_G(\alpha ) . We study the impact of I G ( α ) I_G(\alpha ) on the structure of [ G , α ] [G,\alpha ] . Suppose that each subgroup generated by a subset of I G ( α ) I_G(\alpha ) can be generated by at most r r elements. We show that the rank of [ G , α ] [G,\alpha ] is ( e , r ) (e,r) -bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of I G ( α ) I_G(\alpha ) has odd order, then [ G , α ] [G,\alpha ] has odd order too. Further, if every pair of elements from I G ( α ) I_G(\alpha ) generates a soluble, or nilpotent, subgroup, then [ G , α ] [G,\alpha ] is soluble, or respectively nilpotent.