Solubility of Finite Generalized Frobenius Groups with the Kernel of Odd Index

摘要

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.