Solubility of Finite Generalized Frobenius with the Kernel of Odd Index

Abstract

A finite group $$G$$ is said to be a generalized Frobenius group with kernel$$F$$, if $$F$$ is a proper nontrivial normal subgroup of $$G$$ and for every element $$Fx$$ of prime order of the quotient group $$G/F$$ the coset $$Fx$$ of the group $$G$$ over $$F$$ has only $$p$$-elements for some prime $$p$$ depending on $$x$$. This article considers generalized Frobenius groups with insoluble kernel. We prove that a quotient group of a generalized Frobenius group over its insoluble kernel is a 2-group.