The Number Of Non-abelian Subgroups Of A₃-groups

Suppose that G is a finite p-group. If all subgroups of index pt of G are abelian and at least one subgroup of index pt-1of G is not abelian, then G is called an At-group. This paper gives the triple (μ0,μ1,μ2), where μi(i=0,1,2) denotes the number of Ai-subgroups of index p in A3-groups, respectively. The number of A1-subgroups in A3-groups is also given.