On groups with an involution whose centralizer has finite rank (Q2776803)

Let \(G\) be a locally finite group, and let \(x\) be an involution of \(G\). It is well known that if the centralizer \(C_G(x)\) is finite, then also the groups \([G,x]'\) and \(G/[G,x]\) are finite. Results of this type have also been obtained when certain finiteness conditions are imposed on \(C_G(x)\), showing that also \([G,x]'\) and \(G/[G,x]\) satisfy the same conditions. Adopting this point of view, in this paper the author proves that if \(x\) is an involution of the periodic almost locally soluble group \(G\) and the centralizer \(C_G(x)\) has finite rank, then both groups \([G,x]'\) and \(G/[G,x]\) have finite rank.