Abstract: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 $p$.
- D. Cutler, Another summable $C_\Omega$-group, Proc. Amer. Math. Soc. 26 (1970), 43–44.
- Peter Danchev, Commutative group algebras of $\sigma$-summable abelian groups, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2559–2564. MR 1415581, DOI 10.1090/S0002-9939-97-04052-5
- R. Linton and C. Megibben, Extensions of Totally Projective Groups, Proc. Amer. Math. Soc. 64 (1977), 35–38.
- P. Hill, A summable $C_\Omega$-group, Proc. Amer. Math. Soc. 23 (1969), 428–430.
- P. Hill and W. Ullery, A note on a theorem of May concerning commutative group algebras, Proc. Amer. Math. Soc. 110 (1990), 59–63.
- W. May, Modular group algebras of simply presented abelian groups, Proc. Amer. Math. Soc. 104 (1988), 403–409.
- Paul Hill
- Affiliation: Department of Mathematics, Auburn University, Alabama 36849
- Email: firstname.lastname@example.org
- Received by editor(s): February 18, 1997
- Communicated by: Ken Goodearl
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3133-3135
- MSC (1991): Primary 20K10, 20K07
- DOI: https://doi.org/10.1090/S0002-9939-98-04675-9
- MathSciNet review: 1476137