S. Hadi Jafari | Searcharxiv

arXiv2023

Isoclinism of Algebras With Bracket

We introduce the notion of isoclinism between central extensions in the category of algebras with bracket. We provide several equivalent conditions under which algebras with bracket are isoclinic. We also study the connection between isoclinism and the Schur multiplier of algebras with bracket. It is shown that for finite-dimensional central extensions of algebras with bracket with same dimension, the notion of isoclinism and isomorphism are equivalent. Furthermore, we indicate that all stem covers of an algebra with bracket are isoclinic.

math.RA↗

arXiv2023

Finite split metacyclic groups and their 2-nilpotent multipliers

There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group $G$ be presented as the quotient of a free group $F$ by a normal subgroup $R$. Given a positive integer $c$, the $c$-nilpotent multiplier of the group $G$ is the abelian group $\mathcal M^{(c)}(G)=(R\cap γ_{c+1}(F))/γ_{c+1}(R,F)$, where $γ_1(R,F)=R$, $γ_{c+1}(R,F)=[γ_c(R,F),F]$, and $γ_{c+1}(F)=γ_{c+1}(F,F)$. In particular, $\mathcal{M}^{(1)}(G)$ is the Schur multiplier of $G$. The crucial aspect of the research in to the $c$-nilpotent multipliers of groups includes either establishing their structures, or estimating their sizes and exponents. One reason for studying the $c$-nilpotent multiplier is its relevance to the isologism theory of P. Hall. The study of Schur multiplier of finite metacyclic groups goes back to the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups with the help of their nonabelian tensor squares. In particular, we give a complete description of the triple tensor product, the triple exterior product, and the 2-nilpotent multiplier of such groups.

math.GR↗