Shunkov Groups Saturated with Almost Simple Groups

A group G is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups H in the factor group N_G(H)/H, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set 𝔐 if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in 𝔐. We show that a Shunkov group G which is saturated with groups from the set 𝔐 possessing specific properties, and contains an involution z with the property that the centralizer C_G(z) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in 𝔐. In particular, a Shunkov group G that is saturated with finite almost simple groups and contains an involution z with the property that the centralizer C_G(z) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.