ON THE RANK AND EXPONENT OF THE FIXED POINTS OF COPRIME ACTIONS

Let A be a group acting on a p-group P coprimely. We show that if A centralizes some specified abelian subgroups of P, then A acts trivially on P. As a consequence of this, we obtain that the special rank of C-P(A) is strictly less than that of P unless the action of A on P is trivial. Secondly, we prove that if A acts on a group G coprimely and [G, A] = G, then the exponent of C-G(A)/(C-G(A))' divides vertical bar G : C-G(A)vertical bar.