Finite non-elementary abelian p-groups whose number of subgroups is maximal

Abstract

Assume G is a direct product of Mp(1, 1, 1) and an elementary abelian p-group, where Mp(1, 1, 1) = 〈a, b | ap = bp = cp =1, [a,b]=c,[c,a] = [c,b]=1〉. When p is odd, we prove that G is the group whose number of subgroups is maximal except for elementary abelian p-groups. Moreover, the counting formula for the groups is given.