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

In this paper, we compute the number of fuzzy subgroups of some classes of non-abeilan groups. Explicit formulas are givenfor dihedral groups $D_{2n}$, quasi-dihedral groups $QD_{2^n}$, generalized quaternion groups $Q_{4n}$ and modular $p$-groups $M_{p^n}$.

In this paper we classify fuzzy subgroups of the dihedral group $D_{pqrs}$  for distinct primes  $p$, $q$, $r$ and $s$. This follows similar work we have done on distinct fuzzy subgroups of some dihedral groups.We present formulae for the number of (i) distinct maximal chains of subgroups, (ii) distinct fuzzy subgroups and (iii) non-isomorphic classes of fuzzy subgroups under our chosen equival...

We study here some foundational aspects of the classification problem for torsionfree abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (1n,­), for some n ̄ 1, 2, 3,... . The torsion-free abelian groups of rank% n are the subgroups of (1n,­). For n ̄ 1, that is, the subgroups of (1,­), the isomorphism problem was solved by Baer in the 1930s (see [10...