Minimal presentations for certain metabelian groups

Abstract:

Let G be a finite p-group, $d(G) = \dim {H^1}(G,Z/pZ)$ and $r(G) = \dim {H^2}(G,Z/pZ)$. Then $d(G)$ is the minimal number of generators of G, and we say that G is a member of a class ${\mathcal {G}_p}$ of finite p-groups if G has a presentation with $d(G)$ generators and $r(G)$ relations. The main result is that any outer extension of a finite cyclic p-group by a finite abelian p-group belongs to ${\mathcal {G}_p}$.