On generating a finite group by nilpotent subgroups

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

On the norm of the derived‎ subgroups of all subgroups of a finite group

In this paper‎, ‎we give a complete proof of Theorem 4.1(ii) and a new‎ ‎elementary proof of Theorem 4.1(i) in [Li and Shen‎, ‎On the‎ ‎intersection of the normalizers of the derived subgroups of all‎ ‎subgroups of a finite group‎, ‎ J‎. ‎Algebra, ‎323  (2010) 1349--1357]‎. ‎In addition‎, ‎we also give a generalization of Baer's Theorem‎.

On semi-$Pi$-property of subgroups of finite group

Let $G$ be a group and $H$ a subgroup of $G$‎. ‎ $H$ is said to have semi-$Pi$-property in $G$ if there is a subgroup $T$ of $G$ such that $G=HT$ and $Hcap T$ has $Pi$-property in $T$‎. ‎In this paper‎, ‎investigating on semi-$Pi$-property of subgroups‎, ‎we shall obtain some new description of finite groups‎.

On the planarity of a graph related to the join of subgroups of a finite group

‎Let $G$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎We define an undirected simple graph $Delta(G)$ whose‎ ‎vertices are the proper subgroups of $G$, which are not contained in the‎ ‎Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge‎ ‎if and only if $G=langle H‎ , ‎Krangle$‎. ‎In this paper we classify finite groups with planar graph‎. ‎...

Irredundant Generating Sets of Finite Nilpotent Groups

It is a standard fact of linear algebra that all bases of a vector space have the same cardinality, namely the dimension of the vector space over its base field. If we treat a vector space as an additive abelian group, then this is equivalent to saying that an irredundant generating set for a vector space must have cardinality equal to the dimension of the vector space. The same is not true for...