A note on $\sigma$-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 $p$.