Maximal subgroup growth of a few polycyclic groups

Abstract

We give here the exact maximal subgroup growthof two classes of polycyclic groups. LetGk=⟨x1, x2, . . . , xk|xixjx−1ixjfor alli < j⟩, soGk=Z ⋊(Z ⋊(Z ⋊· · ·⋊ Z)). Then forall integersk⩾2, we calculatemn(Gk), the number of maximalsubgroups ofGkof indexn, exactly. Also, for inőnitely many groupsHkof the form Z2⋊G2, we calculatemn(Hk)exactly.