Bounded Engel and solvable unitary units in group rings
Let \(FG\) be the group algebra of a group \(G\) over a field \(F\) and let \(U(FG)\) be the unit group of the group algebra \(FG\). Consider the classical involution \({\ast}\) on FG, given by \[ {\Big( \sum_{g\in G} \alpha_gg\Big)}^{\ast} = \sum_{g\in G} \alpha_gg^{-1},\\\ \alpha_g \in F. \] The set \(U_n(FG)=\{x\in U(FG): xx^{\ast}=1\}\) is a subgroup of \(U(FG)\) and it is called the group of unitary units of \(FG\). The authors recall that the group algebra \(FG\) is called modular if char\(F=p>0\) and \(G\) has an element of prime order \(p\) and that \(G\) is \(p\)-abelian if \(G\) is a finite \(p\)-group. It is well known when the group \(U(FG)\) is nilpotent [\textit{I. I. Hripta}, Math. Notes 11, 119--124 (1972; Zbl 0258.20005); translation from Mat. Zametki 11, 191--200 (1972); \textit{J. L. Fisher} et al., Proc. Am. Math. Soc. 59, 195--200 (1976; Zbl 0341.16001)] and [\textit{I. I. Khripta}, On the multiplicative group of a group ring. Uzhgorod (Diss.) (1971)]. The main results of the paper are connected with the above indicated results. They are the following. Theorem 1. Let \(F\) be an infinite field of characteristic \(p >\) 2 and \(G\) a group, such that \(FG\) is modular. Then the following are equivalent: 1. \(U_n(FG)\) is bounded Engel and solvable, 2. \(U(FG)\) is nilpotent, and 3. \(G\) is nilpotent and \(p\)-abelian. Theorem 2. Let \(F\) be an algebraically closed field of characteristic different from 2 and \(G\) a group, such that \(FG\) is not modular. Then the following are equivalent: 1. \(U_n(FG)\) is bounded Engel and solvable, 2. \(U(FG)\) is nilpotent, and 3. \(G\) is nilpotent and the torsion elements of \(G\) are central.
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