On units of twisted group algebras (Q1602021)

This paper continues and extends the work of numerous authors on Hartley's problem. Specifically, let \(R\) be an algebra over a field \(K\) and suppose that its group of units \(U(R)\) does not contain a nonabelian free subgroup. The goal is to deduce information on the ring-theoretic structure of \(R\). Of course, it is natural to assume that \(R\) is generated by \(U(R)\), and hence group algebras and twisted group algebras are of particular interest. The main results here are: Theorem 1. Let \(K^t[G]\) be a twisted group algebra of the countable locally finite group \(G\) over the nonabsolute field \(K\) of characteristic \(p>0\). Then \(U(K^t[G])\) contains no nonabelian free subgroup if and only if \(G=PQ\), where \(P\) is a normal Sylow \(p\)-subgroup, \(P\cap Q=1\), and \(K^t[Q]\) is Abelian. Theorem 2. Let \(K^t[G]\) be a twisted group algebra of the locally finite group \(G\) over the infinite field \(K\) of characteristic \(p>0\). The following are equivalent: (1) \(U(K^t[G])\) satisfies a group identity; (2) \(K^t[G]\) satisfies a nonmatrix identity; (3) \(G'\) is a \(p\)-group of bounded period, \(G\) has a normal \(p\)-Abelian subgroup of finite index, and \(K^t[G]\) is stably untwisted; (4) \(U(K^t[G])\) satisfies \((x,y)^{p^k}=1\) for some integer \(k\geq 1\).