A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup HssG contained in H such that $H^{sG}=HT$ and $H\cap T\le H_{\mathit{ssG}},$ where HsG is the intersection of all s-permutable subgroups of G containing H. In this article, we investigate the influence of some nearly s-embedded subgroups of Sylow p-subgroups on the p-nilpotency of finite groups. A number of known results are improved and extended.

A subgroup H of a finite group G is said to be nearly s-embedded in G if G has an s-permutable subgroup T and an s-semipermutable subgroup H_ssG contained in H such that $H^{\mathit{sG}}=\mathit{HT}$ and $H\cap T\le H_{\mathit{ssG}}$, where H^sG is the intersection of all s-permutable subgroups of G containing H. In this paper, we present an extension of a result on p-supersolvability of finite groups under the assumption that maximal subgroups of some Sylow p-subgroups are nearly s-embedded. As applications, we give several new criteria for a finite group to be p-nilpotent and supersolvable.

The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split and threshold. We also classify all finite nilpotent groups whose enhanced power graphs are chordal graphs and cographs. Finally, we give some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and cographs. These results partly answer a question posed by Peter J. Cameron.

A subgroup $ H $ of a finite group $ G $ is called $ s^{\ast} $-normal in $ G $ if some normal subgroup $ T $ of $ G $ satisfies $ H_G\le T $, $ G=HT $, and the quotient $ (H\cap T)/H_G $ has square-free order, where $ H_G $ is the core of $ H $ in $ G $. This paper gave several new criteria for finite groups to be solvable by considering the $ s^{\ast} $-normality of certain $ n $-maximal invariant subgroups under coprime actions, where $ n\in\{1,2,3\} $.