We say that a subgroup $H$ is isolated in group $G$ if for every $x\in G$ we have either H$ or $\langle x\rangle\cap H=1$. In this short note, describe the set of subgroups finite abelian group. The technique used based on an interesting connection between and function sum element orders

Much less known, however, is that there is a description of Aut(G), the automorphism group of G. The first compete characterization that we are aware of is contained in a paper by Ranum [1] near the turn of the last century. Beyond this, however, there are few other expositions. Our goal is to fill this gap, thereby providing a much needed accessible and modern treatment. Our characterization o...

this paper, deals with some equivalence relations in fuzzy subgroups. further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.

This paper, deals with some equivalence relations in fuzzy subgroups. Further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.

Let $G$ be a finite non-abelian $p$-group and $L(G)$ denotes the absolute center of $G$. Also, let $Aut^{L}(G)$ and $Aut_c(G)$ denote the group of all absolute central and the class preserving automorphisms of $G$, respectively. In this paper, we give a necessary and sufficient condition for $G$ such that $Aut_c(G)=Aut^{L}(G)$. We also characterize all finite non-abelian $p$-groups of order $p^...