2022-01-26T11:54:40Z
http://jims.ims.ir/?_action=export&rf=summon&issue=14702
Journal of the Iranian Mathematical Society
JIMS
2020
1
2
Understanding Wall's theorem on dependence of Lie relators in Burnside groups
M.
Vaughan-Lee
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
2020
06
01
129
143
http://jims.ims.ir/article_107524_f75dbbf7321a14000589885d5d7b9665.pdf
Journal of the Iranian Mathematical Society
JIMS
2020
1
2
Approximate biprojectivity and biflatness of some algebras over certain semigroups
H.
Pourmahmood-Aghababa
M. H.
Sattari
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
2020
06
01
145
155
http://jims.ims.ir/article_107698_f1cce5a41ba06bdadbd87490a3330180.pdf
Journal of the Iranian Mathematical Society
JIMS
2020
1
2
A note on factorizations of finite groups
G. M.
Bergman
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
2020
06
01
157
161
http://jims.ims.ir/article_108338_8df75833b842c30981a08daf42636a75.pdf
Journal of the Iranian Mathematical Society
JIMS
2020
1
2
Counting subrings of $mathbb{Z}^n$ of non-zero co-rank
S.
Chimni
G.
Chinta
R.
Takloo-Bighash
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
2020
06
01
163
172
http://jims.ims.ir/article_118868_57f0145a78e287eccee85f1c2bbf2a9d.pdf