In 1954, Kontorovich and Plotkin introduced the concept of a modular chain in a lattice to obtain a lattice-theoretic characterization of the class of torsion-free nilpotent groups. We determine the structure of finite groups with modular chains. It turns out that this class of groups lies strictly between the class of finite groups with lower semimodular subgroup lattice and the projective closure of the class of finite nilpotent groups.

In this paper we consider a prime graph of finite groups. In particular, we expect finite groups with prime graphs of maximal diameter.

We prove that are primitive the finite groups whose normalizers of the Sylow subgroups are primitive. We classify the groups of such class, denoted by $N\mathcal{P}$, and we study the Schunck classes whose boundary is contained in $N\mathcal{P}$. We give also necessary and sufficient conditions in order that the projectors be subnormally embedded.

Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$-subgroup of $G$ with the smallest generator number $d$. There is a set ${ℳ}_d\left(P\right)=\{P_1,P_2,\cdots ,P_d\}$ of maximal subgroups of $P$ such that ${\bigcap }_{i=1}^d{P}_i=\Phi \left(P\right)$. In the present paper, we investigate the structure of a finite group under the assumption that every member of ${ℳ}_d\left(P\right)$ is either $s$-permutably embedded or weakly $s$-permutable in $G$ to give criteria for a group to be $p$-supersolvable or $p$-nilpotent.

A subgroup $H$ of a finite group $G$ is said to be SS-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is S-quasinormal in $K$. We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results.

Suppose G is a finite group and H is a subgroup of G. H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup $H_{se}$ of G contained in H such that G = HT and $H\cap T\le H_{se}$; H is called weakly s-supplemented in G if there is a subgroup T of G such that G = HT and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of the existence of s-permutably embedded and weakly s-supplemented...

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. Subgroup $H$ is said to be weakly $ℳ$-supplemented in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G=HB$, and (2) if $H_1/H_G$ is a maximal subgroup of $H/H_G$, then $H_{1}B=B{H}_1<G$, where $H_G$ is the largest normal subgroup of $G$ contained in $H$. We fix in every noncyclic Sylow subgroup $P$ of $G$ a subgroup $D$ satisfying $1<\left|D\right|<\left|P\right|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $\left|H\right|=\left|D\right|$ is weakly $ℳ$-supplemented in $G$. Some recent results are generalized.

We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for ${SL}_2$ which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.