ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP

Abstract

We construct, for the first time, various types of specific non-special finite p-groups having abelian automorphism group. More specifically, we construct groups G with abelian automorphism group such that γ2(G) < Z(G) < Φ(G), where γ2(G), Z(G) and Φ(G) denote the commutator subgroup, the center and the Frattini subgroup of G respectively. For a finite p-group G with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) Z(G) = Φ(G) is elementary abelian; (ii) γ2(G) = Φ(G) is elementary abelian, where p is an odd prime. We construct examples to show the existence of groups G with elementary abelian automorphism group for which exactly one of the above two conditions holds true.