A note on σ-summable groups

We answer questions raised by P. Danchev in a recent paper in these Proceedings. It is shown that a sigma-summable abelian p-group is not determined by its socle, that is, two such groups can have isometric socles without being isomorphic. It is also demonstrated that sigma-summability plays essentially no role in regard to the question of whether or not V(G)/G is totally projective, where V(G) denotes the group of normalized units of the group algebra F(G) with F being a perfect field of characteristic p.