A finite group G is called 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 p in the quotient group G/F, the coset Fx of G consists of p-elements. We study generalized Frobenius groups with an insoluble kernel F. It is proved that F has a unique non- Abelian composition factor, and that this factor is isomorphic to \( {\mathrm{L}}_2\left({3}^{2^{\mathrm{l}}}\right) \) for some natural number l. Moreover, we look at a (not necessarily finite) group generated by a coset of some subgroup consisting solely of elements of order three. It is shown that such a group contains a nilpotent normal subgroup of index three.

如果F是G的一个适当的非平凡正规子群，则有限群G称为带有核F的广义Frobenius群，并且对于商群G / F中素数为p的每个元素Fx，G的陪集Fx由p-元素。我们研究了具有不可溶核F的广义Frobenius群。证明了F具有唯一的非阿贝尔组成因子，并且该因子与\（{\ mathrm {L}} _ 2 \ left（{3} ^ { 2 ^ {\ mathrm {l}}} \ right）\）中的某个自然数l。而且，我们看一个（仅是有限的）组，该组是由仅由三阶元素组成的某个子组的陪集生成的。结果表明，这样的组包含索引为3的幂等正常子组。