Journal of the Iranian Mathematical SocietyJournal of the Iranian Mathematical Society
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Feed provided by Journal of the Iranian Mathematical Society. Click to visit.Understanding Wall's theorem on dependence of Lie relators in Burnside groups
http://jims.ims.ir/article_107524_14674.html
‎G.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.Sun, 31 May 2020 19:30:00 +0100Approximate biprojectivity and biflatness of some algebras over certain semigroups
http://jims.ims.ir/article_107698_14674.html
‎We 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.Sun, 31 May 2020 19:30:00 +0100A note on factorizations of finite groups
http://jims.ims.ir/article_108338_14674.html
In 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.Sun, 31 May 2020 19:30:00 +0100Sanov's theorem on Lie relators in groups of exponent $p$
http://jims.ims.ir/article_110856_0.html
‎I give a proof of Sanov's theorem that the Lie relators of weight at most‎ ‎$2p-2$ in groups of exponent $p$ are consequences of the identity $px=0$ and‎ ‎the $(p-1)$-Engel identity‎. ‎This implies that the order of the class $2p-2$‎ ‎quotient of the Burnside group $B(m,p)$ is the same as the order of the class‎ ‎$2p-2$ quotient of the free $m$ generator $(p-1)$-Engel Lie algebra over‎ ‎GF$(p)$‎. ‎To make the proof self-contained I have also included a derivation of‎ ‎Hausdorff's formulation of the Baker Campbell Hausdorff formula‎.Sun, 19 Jul 2020 19:30:00 +0100