We consider the question: When do two finite abelian groups have isomorphic lattices of characteristic subgroups? An explicit description of the characteristic subgroups of such groups enables us to give a complete answer to this question in the case where at least one of the groups has odd order. An “exceptional” isomorphism, which occurs between the lattice of characteristic subgroups of Zp ×...

Y. Li gave a characterization of the class of finite soluble groups in which every subnormal subgroup is normal by means of NE -subgroups: a subgroup H of a group G is called an NE -subgroup of G if NG(H) ∩ H = H. We obtain a new characterization of these groups related to the local Wielandt subgroup. We also give characterizations of the classes of finite soluble groups in which every subnorma...

Let S be a 2-group. The rank (normal rank) of S is the maximal dimension of an elementary abelian subgroup (a normal elementary abelian subgroup) of S over Z2. The purpose of this article is to determine the rank and normal rank of S, where S is a Sylow 2-subgroup of the classical groups of odd characteristic.

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

A finite p-group S is said to be of supersoluble type if every fusion system over supersoluble. The main aim this paper characterise the p-groups type. Abelian and metacyclic are completely described. Furthermore, we show that Sylow p-subgroups a simple group must cyclic.