The Number Of Minimal Non-abelian Subgroups In A_t-groups

If all subgroups of index pt of a finite p-group G are abelian and at least one subgroup of index pt-1of G is not abelian, then G is called an At-group. In this paper, it is given that the number of minimal non-abelian subgroups in the direct product of an At-group and a cyclic group, an elementary abelian group, respectively. And it is also given that the upper and lower bound of numbers of minimal non-abelian subgroups in the direct product of an At-group and an abelian group. In addition, we give an estimate for the range of the minimum value of numbers of minimal non-abelian subgroups in an At-groups.