On groups with involutions saturated by finite Frobenius groups

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 $F$.