Characterization of finite abelian p-groups satisfying P_{p-1}

Determine whether every noncyclic finite abelian p-group satisfies property P_{p-1}, equivalently, whether a finite abelian p-group G satisfies P_{p-1} if and only if G is noncyclic.

Background

The paper defines property P_k for a finite group G by requiring that, for every collection of k automorphisms of G, there is a single bijection f of G such that each map g ↦ β_i(g)f(g) is also a bijection. The authors prove that every finite group satisfies P_k whenever k is smaller than p−1, where p is the smallest prime divisor of the group order.

For finite abelian p-groups, the authors obtain the stronger result that the common bijection f can be chosen to be an automorphism when k<p−1. They also prove that cyclic p-groups do not satisfy P_{p−1}, while noncyclic homocyclic p-groups do satisfy P_{p−1}. These results motivate the unresolved characterization of all noncyclic finite abelian p-groups at the threshold k=p−1. The authors note that, using characteristic subgroups and their extension result, the question reduces to groups of the form Z_p^n × Z_{p^e}, with n,e≥1, beyond the already settled homocyclic base case.