Groups with all elements of finite length (Q1812158)

Let \(G\) be a group and let \(x\in G\). An `\(x\)-basis' is a sequence of conjugates \((x^{g_1},\dots,x^{g_n})\) such that \(x^G=\langle x^{g_1},\dots,x^{g_n}\rangle\) and \(x^{g_i}\notin\langle x^{g_1},\dots,x^{g_{i-1}}\rangle\) for \(i=2,\dots,n\). The length of a maximal \(x\)-basis (if one exists) is written \(\ell_G(x)\). A group \(G\) is called an `FL-group' (``finite length'') if \(\ell_G(x)\) is finite for all \(x\in G\). Also \(G\) is a `BFL-group' (``bounded finite length'') if \(\ell_G(x)\) is boundedly finite for all \(x\in G\). The authors establish connections between these classes and the classes of FC-groups (finite conjugacy classes) and BFC-groups (boundedly finite conjugacy classes). Specifically they prove that for locally graded groups FC and FL coincide, as do BFC and BFL. (The structures of FC-groups and BFC-groups are of course well-known).