Groups with finitely imbedded involution (Q1814621)

Main theorem: If \(G\) is a group with finitely embedded involution \(i\), \(B = \langle i^ G\rangle\), \(R = [B,G]\), if \(Z\) is the subgroup generated by all 2-elements of \(R\), and further if \(\langle i, i^ g \rangle\) is finite for all \(g\) in \(G\), then \(B\), \(R\) and \(Z\) are normal subgroups of \(G\) and either (i) \(B\) is finite or (ii) \(B\) is locally finite, a split extension of \(R\) by \(\langle i \rangle\), and \(Z\) is a finite extension of a divisible abelian 2-group \(A_ 2\) with minimal condition such that \(ici = c^{-1}\) for all \(c\) in \(A_ 2\).