The influence of ℳ-supplemented subgroups on the structure of chief factors of finite groups
In this paper, we introduce the concept of weakly [Formula: see text]-hypercyclically embedded subgroups, and investigate the influence of [Formula: see text]-supplemented subgroups on the structure of chief factors of finite groups. First, we find a connection between [Formula: see text]-supplemented subgroups and normally embedded subgroups. With the help of this connection, we give some criteria for (weakly) [Formula: see text]-hypercyclically embeddability of normal subgroups of finite groups by using fewer [Formula: see text]-supplemented [Formula: see text]-subgroups with given order. In particular, we not only simplify, but also improve the Main Theorem of [L. Miao and J. Zhang, On a class of non-solvable groups, J. Algebra 496 (2018) 1–10]. Finally, we point out that for a [Formula: see text]-subgroup [Formula: see text] of [Formula: see text], the concept of [Formula: see text]-embedded subgroups coincides with the concept of [Formula: see text]-supplemented subgroups.