Projective bases of division algebras and groups of central type.
The paper under review continues the study, started by the first two authors, of those twisted group algebras over a field \(k\), which are in fact central division algebras. They have proved [see Isr. J. Math. 121, 173-198 (2001; Zbl 0978.16027)] that the groups corresponding to these algebras are nilpotent, and also, have reduced the presentation of the algebras considered to the description of those finite \(p\)-groups \(P\) for which the twisted group algebra \(k^\alpha P\) (where \(\alpha\) is a \(2\)-cocycle) is of the type dealt with. The present paper provides a short list of such groups, and announces (to appear in a forthcoming paper by the third author) that \(P\) has the required property if and only if it is isomorphic to a direct product of groups from the list. Also, it completes the proof of the assertion that the algebras discussed are isomorphic to tensor products of cyclic division \(k\)-algebras.
scientific article; zbMATH DE number 2174715