Symmetric units satisfying a group identity. (Q876407)

Let \(U(KG)\) be the group of units of the group algebra \(KG\) of a locally finite group \(G\) over a field \(K\) of \(\text{char}(K)\neq 2\). Let \(\varphi\colon KG\to KG\) be the \(K\)-linear extension of an anti-automorphism \(\varphi\) of order \(2\) on \(G\) and set \(S_\varphi(KG)=\{u\in U(KG)\mid\varphi(u)=u\}\). In the present paper the authors investigate the properties of \(S_\varphi(KG)\). The problem goes back to \textit{V.~Bovdi, L.~G.~Kovács and S.~K.~Sehgal} [Commun. Algebra 24, No. 3, 803-808 (1996; Zbl 0846.20007)] and \textit{V.~Bovdi} [Commun. Algebra 29, No. 12, 5411-5422 (2001; Zbl 0994.16023)] who first began to study the question when the symmetric units (under the classical involution) in a group algebra form a subgroup. As a possible generalization of these investigations it is a natural question: when do the symmetric units satisfy a group identity in a group algebra? For the case when \(\text{char}(K)>2\) and \(G\) is a torsion group this problem was solved by \textit{A.~Giambruno, S.~K.~Sehgal and A.~Valenti}, [Manuscr. Math. 96, No. 4, 443-461 (1998; Zbl 0910.16015)], and for an arbitrary group \(G\) by \textit{V.~Bovdi}, [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 22, No. 2, 149-159 (2006)]. In the present paper the authors prove that if \(S_\varphi(KG)\) satisfies a group identity then \(KG\) satisfies a polynomial identity. Moreover, in case when the prime radical of \(KG\) is nilpotent they give a description of \(G\) for which \(S_\varphi(KG)\) satisfies a group identity.