On Groups with Involutions Saturated by Finite Frobenius Groups

Abstract

We study the mixed and periodic groups with involutions and finite elements which are saturated by finite Frobenius groups. We prove that a group $G$ of $2$-rank 1 of even order greater than 2 splits into the direct product of a periodic abelian group $F$ and the centralizer of an involution; moreover, each maximal periodic subgroup in $G$ is a Frobenius group with kernel $F$. We characterize one class with the saturation condition. We prove that a group of $2$-rank greater than 1 with finite elements of prime orders is a split extension of a periodic group $F$ by a group $H$ in which all elements of prime orders generate a locally cyclic group; moreover, every element in $F$ with every element of prime order in $H$ generates a finite Frobenius group. Under the condition of the triviality of the local finite radical, we determine some properties of the subgroup $F$.