Journal of Algebraic Systems
Volume:4 Issue: 1, Summer - Autumn 2016

Let (G;N) be a pair of groups. In this article, first we con- struct a relative central extension for the pair (G;N) such that special types of covering pair of (G;N) are homomorphic image of it. Second, we show that every perfect pair admits at least one covering pair. Finally, among extending some properties of perfect groups to perfect pairs, we characterize covering pairs of a perfect pair (G;N) under some extra assumptions.

Let R be a commutative ring with identity and M be a unitary R-module. In this paper we generalize the concept multiplicatively closed subset of R and we study some properties of these genaralized subsets of M. Among the many results in this paper, we generalize some well-known theorems about multiplicatively closed subsets of R to these generalized subsets of M. Also we show that some other well-known results about multiplicatively closed subsets of R are not valid for these generalized subsets of M.

Let H be a subgroup of a finite group G. We say that H is SS-semipermutable in G if H has a supplement K in G such that H permutes with every Sylow subgroup X of K with (jXj; jHj) = 1. In this paper, the Structure of SS-semipermutable subgroups, andfinite groups in which SS-semipermutability is a transitive relation are described. It is shown that a finite solvable group G is a PST-group if and only if whenever H  K are two p-subgroups of G, H is SS-semipermutable in NG(K).

The prime graph of a finite group $G$ is denoted by $ga(G)$. A nonabelian simple group $G$ is called quasirecognizable by prime graph, if for every finite group $H$, where $ga(H)=ga(G)$, there exists a nonabelian composition factor of $H$ which is isomorphic to $G$. Until now, it is proved that some finite linear simple groups are quasirecognizable by prime graph, for instance, the linear groups $L_n(2)$ and $L_n(3)$ are quasirecognizable by prime graph. In this paper, we consider the quasirecognition by prime graph of the simple group $L_n(5)$.

Let R be a commutative ring. An R-module M is called co-multiplication provided that for each submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper we show that co-multiplication modules are a generalization of strongly duo modules. Uniserial modules of finite length and hence valuation Artinian rings are some distinguished classes of co-multiplication rings. In addition, if R is a Noetherian ring, then R is a strongly duo ring if and only if R is a co-multiplication ring. We also show that J-semisimple strongly duo rings are precisely semisimple rings. Moreover, if R is a perfect ring, then uniserial R modules are co-multiplication of finite length modules. Finally, we show that Abelian co-multiplication groups are reduced and co-multiplication Z-modules(Abelian groups)are characterized.

In this paper we study the signed Roman domination number of the join of graphs. Specially, we determine it for the join of cycles, wheels, fans and friendship graphs.

Let $R$ be a commutative Noetherian ring and let $fa$, $fb$ be two ideals of $R$ such that $R/({fa})$ is Artinian. Let $M$, $N$ be two finitely generated $R$-modules. We prove that $H_{fb}^j(H_{fa}^t(M,N))$ is Artinian for $j=0,1$, where $t=inf{iin{mathb {N}_0}: H_{fa}^i(M,N)$ is not finitely generated $}$. Also, we prove that if $DimSupp(H_{fa}^i(M,N))leq 2$, then $H_{fb}^ (H_{fa}^i(M,N))$ is Artinian for all $i$. Moreover, we show that if $dim N=d$, then $H {fb}^j(H_{fa}^{d-1}(N))$ is Artinian for all $jgeq 1$.