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5 votes
1 answer
349 views

Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity

Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $\mathrm{GL}_n(\mathbb{Z})$, $\forall n\geq 0$. I have ...
Guest's user avatar
  • 51
3 votes
0 answers
70 views

Admissibility of Ulm's invariants

Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define $$G_{\alpha}=pG_{\beta}.$$ If $\alpha$ is a limit ...
Nini's user avatar
  • 31
4 votes
0 answers
219 views

Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$

EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect. Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
Libli's user avatar
  • 7,300
3 votes
0 answers
71 views

Automorphisms of matrix algebras and Picard group

This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE). Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
GreginGre's user avatar
  • 1,766
4 votes
1 answer
277 views

Is there a good notion of kernels of quadratic forms on abelian groups?

Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the ...
Bipolar Minds's user avatar
5 votes
1 answer
523 views

Is there a non-split algebraic torus (over a finite field) satisfying the following properties?

Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties? $T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
Dimitri Koshelev's user avatar
2 votes
2 answers
265 views

Commuting nilpotent matrices and conjugation isomorphisms

Trying to study isomorphism classes of certain commutative Artinian $\mathbb{C}$-algebras I was lead to the following problem about matrices. Suppose you have a (non-zero) nilpotent matrix $A\in M_n(\...
amateur's user avatar
  • 385
0 votes
1 answer
454 views

Conjugacy in the quaternion group

Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
Gautam's user avatar
  • 1,703
1 vote
0 answers
126 views

Algebraic structures on graphs

There are many algebraic structures linked to graphs. For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs. Does there exist any survey paper which characterizes all the ...
Charlotte's user avatar
  • 444
2 votes
1 answer
413 views

$2$-adic valuation on $\mathbb Q (\sqrt{-15})$

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that $-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ ...
usermath's user avatar
  • 243
19 votes
1 answer
4k views

How should I think about the module of coinvariants of a $G$-module?

Let $G$ be a group, $M$ a $G$-module, then the group of coinvariants is the module $M_G := M/I_GM$, where $I_G$ is the kernel of the augmentation map $\epsilon : \mathbb{Z}G\rightarrow \mathbb{Z}$. ...
stupid_question_bot's user avatar
8 votes
1 answer
353 views

$E_n(\ell^\infty)=SL_n(\ell^\infty)$?

Let $R$ be a commutative unital ring $R$ with unit element $1$. For $n\in \mathbb{N}=\{1,2,3,\cdots\}$, let $SL_n(R)$ be the group of all $n\times n$ matrices with entries from $R$ having ...
KevinC's user avatar
  • 81
15 votes
1 answer
679 views

Submodules of $({\mathbb Z}/6{\mathbb Z})^n$ intersecting $\{0,1\}^n$ trivially

$\newcommand{\F}{{\mathbb F}}$ $\newcommand{\Z}{{\mathbb Z}}$ Suppose that $\F$ is a finite field of prime order $p:=|\F|$, and let $n$ be a positive integer. I consider the regime where $\F$ is ...
Seva's user avatar
  • 23k
3 votes
1 answer
891 views

Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...
Jérémy Blanc's user avatar