Free symmetric and unitary pairs in group algebras with involution (Q332621)

Let \(A\) be an algebra over a field \(k\). Endowing it with a \(k\)-involution \(*\) allows us to speak about symmetric, and unitary elements, satisfying respectively \(a=a^{*}\), and \(aa^{*}=a^{*}a\). A pair of units of \(A\) if free, if it generates a free subgroup of the group of units of \(A\). Now, if \(A=kG\) is a group algebra, the map \(g\mapsto g^{-1}\) naturally extends to an involution of \(A\). The article gives necessary and sufficient conditions for the existence of free pairs of symmetric or unitary units of \(kG\), for non-abelian locally finite groups \(G\), and non-absolute fields (i.e., fields which are not algebraic over a finite subfield) \(k\) of characteristic \(\neq 2\).