Iranian Mathematical Society
Journal of the Iranian Mathematical Society
2717-1612
1
2
2020
12
01
Understanding Wall's theorem on dependence of Lie relators in Burnside groups
129
143
EN
M.
Vaughan-Lee
Christ Church, University of Oxford,
Oxford, OX1 1DP, England.
michael.vaughan-lee@chch.ox.ac.uk
10.30504/jims.2020.107524
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 $n\neq1\operatorname{mod}(p-1)$, 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.
Lie relators,Burnside groups,Wall's theorem
https://jims.ims.ir/article_107524.html
https://jims.ims.ir/article_107524_f75dbbf7321a14000589885d5d7b9665.pdf
Iranian Mathematical Society
Journal of the Iranian Mathematical Society
2717-1612
1
2
2020
12
01
Approximate biprojectivity and biflatness of some algebras over certain semigroups
145
155
EN
H.
Pourmahmood-Aghababa
Department of Mathematics, University of Tabriz, Tabriz, Iran.
pourmahmood@gmail.com
M. H.
Sattari
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
sattari@azaruniv.edu
10.30504/jims.2020.107698
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.
Banach algebras,approximate biprojectivity,biflatness,inverse semigroups
https://jims.ims.ir/article_107698.html
https://jims.ims.ir/article_107698_f1cce5a41ba06bdadbd87490a3330180.pdf
Iranian Mathematical Society
Journal of the Iranian Mathematical Society
2717-1612
1
2
2020
12
01
A note on factorizations of finite groups
157
161
EN
G. M.
Bergman
0000-0003-4027-7293
Department of Mathematics,
University of California
Berkeley, CA 94720-3840
USA.
gbergman@math.berkeley.edu
10.30504/jims.2020.108338
In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group $G$ and a factorization $\mathrm{card}(G)= n_1\ldots n_k,$ one can always find subsets $A_1,\ldots,A_k$ of $G$ with $\mathrm{card}(A_i)=n_i$ such that $G=A_1\ldots A_k;$ equivalently, such that the group multiplication map $A_1\times\ldots\times A_k\to G$ 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.
Factorization of a finite group,Product of subsets
https://jims.ims.ir/article_108338.html
https://jims.ims.ir/article_108338_8df75833b842c30981a08daf42636a75.pdf
Iranian Mathematical Society
Journal of the Iranian Mathematical Society
2717-1612
1
2
2020
12
01
Counting subrings of $\mathbb{Z}^n$ of non-zero co-rank
163
172
EN
S.
Chimni
Department of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.
sarthakchimni@gmail.com
G.
Chinta
Department of Mathematics, The City College of New York, New York, NY 10031.
gchinta@ccny.cuny.edu
R.
Takloo-Bighash
0000-0002-5340-2412
Department of Mathematics, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S Morgan St (M/C 249), Chicago, IL 60607.
rtakloo@gmail.com
10.30504/jims.2020.238412.1020
In 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 $r$.
$mathbb{Z^n}$,subrings,Stirling numbers of the second kind
https://jims.ims.ir/article_118868.html
https://jims.ims.ir/article_118868_57f0145a78e287eccee85f1c2bbf2a9d.pdf