Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121220200601Understanding Wall's theorem on dependence of Lie relators in Burnside groups12914310752410.30504/jims.2020.107524ENM.Vaughan-LeeChrist Church, University of Oxford,
Oxford, OX1 1DP, England.Journal Article20200325G.E. Wall [J. Algebra 104 (1986), no. 1, 1--22; Lecture Notes in Mathematics, pp. 191--197, 1456, Springer-Verlag, Berlin, 1990] gave two different proofs of a remarkable result about the multilinear Lie relators satisfied by groups of prime power exponent $q$. He showed that if $q$ is a power of the prime $p$, and if $f$ is a multilinear Lie relator in $n$ variables where $nneq1operatorname{mod}(p-1)$, then $f=0$ is a consequence of multilinear Lie relators in fewer than $n$ variables. For years I have struggled to understand his proofs, and while I still have not the slightest clue about his proof in [J. Algebra 104 (1986), no. 1, 1--22], I finally have some understanding of his proof in [Lecture Notes in Mathematics, pp. 91--197, 1456, Springer-Verlag, Berlin, 1990]. In this note I offer my insights into Wall's second proof of this theorem.Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121220200601Approximate biprojectivity and biflatness of some algebras over certain semigroups14515510769810.30504/jims.2020.107698ENH.Pourmahmood-AghababaDepartment of Mathematics, University of Tabriz, Tabriz, Iran.M. H.SattariDepartment of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.Journal Article20190715We investigate (bounded) approximate biprojectivity of $l^1(S)$ for uniformly locally finite inverse semigroups. As a consequence, we show that when $S=mathcal{M}(G, I)$ is the Brandt inverse semigroup, then $l^1(S)$ is (boundedly) approximately biprojective if and only if $G$ is amenable. Moreover, we study biflatness and (bounded) approximate biprojectivity of the measure algebra $M(S)$ of a topological Brandt semigroup.Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121220200601A note on factorizations of finite groups15716110833810.30504/jims.2020.108338ENG. M.BergmanDepartment of Mathematics,
University of California
Berkeley, CA 94720-3840
USA.0000-0003-4027-7293Journal Article20200518In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group $G$ and a factorization $mathrm{card}(G)= n_1ldots n_k,$ one can always find subsets $A_1,ldots,A_k$ of $G$ with $mathrm{card}(A_i)=n_i$ such that $G=A_1ldots A_k;$ equivalently, such that the group multiplication map $A_1timesldotstimes A_kto G$ is a bijection. We show that for $G$ the alternating goup on $4$ elements, $k=3,$ and $(n_1,n_2,n_3) = (2,3,2),$ the answer is negative. We then generalize some of the tools used in our proof, and note a related open question.Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121220200601Counting subrings of $mathbb{Z}^n$ of non-zero co-rank16317211886810.30504/jims.2020.238412.1020ENS.ChimniDepartment of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.G.ChintaDepartment of Mathematics, The City College of New York, New York, NY 10031.R.Takloo-BighashDepartment of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.0000-0002-5340-2412Journal Article20200707In this paper we study subrings of $mathbb{Z^{n+k}}$ of co-rank $k$. We relate the number of such subrings $R$ with torsion subgroup $(mathbb{Z^{n+k}}/R)_{rm{tor}}$ of size $r$ to the number of full rank subrings of $mathbb{Z^n}$ of index $r$.