This article demonstrates that finite solvable groups have an LC-nilpotent series if they lack specific 2-Frobenius sections, suggesting a new variety in group theory.
For a finite group G, let LC(G) be the subgroup generated by elements x such that, for all y∈G and all integers n, the order of xny divides the least common multiple of the orders of x and y. This subgroup is a nilpotent characteristic subgroup of G. In this article, among other results, we show that a finite solvable group G admits an LC-nilpotent series if and only if G does not contain any 2-Frobenius section of type (p,q,p). As a consequence of this theorem, we conclude that the algebraic system consisting all LC-nilpotent groups forms a variety. Finally, we answer to Question 3.7 from [Citation1] in a more general case.
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Amiri et al. (2026) studied this question.
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