Commutative twisted group algebras (Q2762950)

The authors announce some results on the \(p\)-Sylow subgroup \(S(R_tG)\) of the unit group \(U(R_tG)\) of a commutative twisted group algebra \(R_tG\) of an Abelian group \(G\) over a unitary commutative ring \(R\). Although \(U(R_tG)\) and \(S(R_tG)\) are important objects in the theory of commutative twisted group rings, their description, up to isomorphism, is only given in a few partial cases.NEWLINENEWLINENEWLINEThe first result of the paper generalizes the well known fact that, for a free Abelian group \(G\), the \(R\)-algebra \(R_tG\) is isomorphic to the ordinary group algebra \(RG\) and establishes a similar result when \(G\) is a \(p\)-group and \(R^*\) is a \(p\)-divisible group and when \(R^*\) is a divisible group. The second result gives that if \(R_tG\) and \(R_{t_1}H\) are \(R\)-isomorphic, then the group algebras \(KG\) and \(KH\) are isomorphic for a suitable algebraically closed field \(K\). Then the authors describe the Ulm-Kaplansky invariants of \(S(R_tG)\) in the important case when \(\text{char }R=p\) and the \(p\)-component \(G_p\) of \(G\) is trivial. Further, they introduce the notion of unitly \(p\)-perfect ring as a ring \(R\) of prime characteristic \(p\) and with a \(p\)-divisible group \(R^*\). The authors calculate the Ulm-Kaplansky invariants and the maximal divisible subgroups of \(S(R_tG)\) for unitly \(p\)-perfect rings \(R\). As a corollary they obtain that for a \(p\)-group \(G\) and a unitly \(p\)-perfect ring \(R\), the \(R\)-isomorphism of \(R_tG\) and \(R_{t_1}H\) implies the \(R\)-isomorphism of \(RG\) and \(RH\).