Understanding Wall's theorem on dependence of Lie relators in Burnside groups
M.
Vaughan-Lee
Christ Church, University of Oxford,
Oxford, OX1 1DP, England.
author
text
article
2020
eng
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.
Journal of the Iranian Mathematical Society
Iranian Mathematical Society
2717-1612
1
v.
2
no.
2020
129
143
http://jims.ims.ir/article_107524_f75dbbf7321a14000589885d5d7b9665.pdf
dx.doi.org/10.30504/jims.2020.107524
Approximate biprojectivity and biflatness of some algebras over certain semigroups
H.
Pourmahmood-Aghababa
Department of Mathematics, University of Tabriz, Tabriz, Iran.
author
M. H.
Sattari
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
author
text
article
2020
eng
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.
Journal of the Iranian Mathematical Society
Iranian Mathematical Society
2717-1612
1
v.
2
no.
2020
145
155
http://jims.ims.ir/article_107698_f1cce5a41ba06bdadbd87490a3330180.pdf
dx.doi.org/10.30504/jims.2020.107698
A note on factorizations of finite groups
G. M.
Bergman
Department of Mathematics,
University of California
Berkeley, CA 94720-3840
USA.
author
text
article
2020
eng
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.
Journal of the Iranian Mathematical Society
Iranian Mathematical Society
2717-1612
1
v.
2
no.
2020
157
161
http://jims.ims.ir/article_108338_8df75833b842c30981a08daf42636a75.pdf
dx.doi.org/10.30504/jims.2020.108338
Counting subrings of $\mathbb{Z}^n$ of non-zero co-rank
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.
author
G.
Chinta
Department of Mathematics, The City College of New York, New York, NY 10031.
author
R.
Takloo-Bighash
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.
author
text
article
2020
eng
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 $r$.
Journal of the Iranian Mathematical Society
Iranian Mathematical Society
2717-1612
1
v.
2
no.
2020
163
172
http://jims.ims.ir/article_118868_57f0145a78e287eccee85f1c2bbf2a9d.pdf
dx.doi.org/10.30504/jims.2020.238412.1020