# Tagged Questions

**2**

votes

**0**answers

66 views

### Semidirect products of semigroups [closed]

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$.
A function ...

**3**

votes

**1**answer

366 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 ...

**5**

votes

**3**answers

416 views

### Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$

Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of ...

**3**

votes

**0**answers

162 views

### Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...

**2**

votes

**1**answer

179 views

### Finite-index free subgroups in lattices and matrix rings

It is a theorem of Selberg that a lattice $\Gamma$ in a linear group has a torsion-free subgroup of finite index. Page 64 in 'Introduction to Arithmetic Groups' by Dave Morris asserts these can be ...

**2**

votes

**1**answer

89 views

### Example involving partially ordered Abelian groups

Definition 1:
Let $(G,\leq)$ be a nonzero partially ordered Abelian group with order unit $u$. (Recall that $u\in G$ is a order unit if, for every $g\in G$, there exists $N\in\mathbb N$ such that ...

**6**

votes

**2**answers

339 views

### Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$

The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian $\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for ...

**8**

votes

**1**answer

473 views

### Homomorphisms from powers of Z to Z

I believe it is known that if I is a set of non-measurable cardinality, then any homomorphism $Z^I\to Z$ factors through a finite power. Here $Z$ is the group of integers. Can anyone give a ...

**1**

vote

**1**answer

259 views

### Representations over $\mathbb{Z}_p$

Hi,
I would like to work with indecomposable representations over a commutative ring and start with $R=\mathbb{Z}_p$
Notations: $G$ a finite group, $S$ a subgroup, $R=\mathbb{Z}_p$ the p-adic ring, ...

**4**

votes

**3**answers

288 views

### Spectrum and scheme of the commutative group-algebra of an abelian group.

The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? ...

**3**

votes

**3**answers

370 views

### Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...

**1**

vote

**2**answers

185 views

### My output of a group and inverse-closed subset in MAGMA is no longer inverse-closed when entered as input to GAP.

In MAGMA, I input the following:
G:=SmallGroup(20,3);
G;
E:=[xx:xx in G];
S:=[E[6],E[7],E[13],E[20]];
S;
S[1]^2;
S[2]^2;
S[3]*S[4];
This gives the output:
GrpPC : ...

**3**

votes

**0**answers

190 views

### a normal form for matrices over Z[x]/(x^2-1) ?

We are discussing, offline, modules over the $\mathbb{Z}$-group ring of the cyclic group of order 2, which is probably better known as the quotient ring $R=\mathbb{Z}[t]/(t^2-1)$.
Is there any way to ...

**1**

vote

**1**answer

150 views

### fixed point scheme in caracteristic p

Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k.
Let G a finite group acting on X and suppose that his order is divisible by the caracteristic ...

**4**

votes

**2**answers

272 views

### When is a torsionfree subgroup contained in a torsionfree direct summand?

Let $F$ be a torsionfree subgroup of a commutative group $G$. Are there nontrivial conditions known under which there exists a torsionfree direct summand of $G$ containing $F$?
I would already be ...

**19**

votes

**2**answers

1k views

### Criteria for irreducibility of polynomial

If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best

**4**

votes

**1**answer

188 views

### Heisenberg-type groups over rings with involution

Hello everyone!
In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:
Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...

**43**

votes

**2**answers

2k views

### a categorical Nakayama lemma?

There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...

**13**

votes

**2**answers

474 views

### A space of ideals

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...

**15**

votes

**2**answers

568 views

### Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?

**4**

votes

**1**answer

521 views

### Free and surface groups cohomology

What is a good reference for results on cohomology of finite rank free groups and surface groups with group ring coefficients?
I am interested in the case when the group acts on its group ring via ...

**3**

votes

**2**answers

383 views

### How to make a function depending on some operation?

Let $S$ be an infinite set and $f:S\times S\rightarrow\mathbb R$ be bounded.
Question: Are there simple hypotheses on $f$ such that there is a commutative and associative operation $\cdot$ on $S$ ...

**2**

votes

**0**answers

163 views

### How many generators for rings of partially symmetric polynomials?

Let $k$ be a field, $n$ a positive integer. The group $S_n$ acts on $R_n=k[x_1,\dots,x_n]$ by permuting indices, and $\mathcal{S}_n=R_n^{S_n}=k[s_1,\dots,s_n]$ where the $s_i$'s are the usual ...

**1**

vote

**0**answers

257 views

### Size of an abelian permutation group with generators of order 2 [closed]

Let $g_1, \ldots, g_k$ be distinct permutations on a set $\Omega$. Suppose that $G = \langle g_1, \ldots, g_k \rangle$ is an abelian permutation group with only elements of order at most 2. Is it ...

**0**

votes

**1**answer

267 views

### Elements in terms of other in an abelian permutation group

Let $G$ be an abelian transitive permutation group acting on $\Omega$. Let $\{g_1, \ldots, g_n\}$ be elements of $G$ such that, $\forall i \not = j$, there is no $k \in \mathbb{N}$ such that $g_i = ...

**4**

votes

**0**answers

638 views

### $Ext$ functor, filtered complexes and spectral sequences

Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...

**5**

votes

**2**answers

336 views

### Invariant means on commutative magmas

It is a very standard fact that commutative semigroups admit an invariant mean and the proof basically relies on Markov-Kakutani fixed point theorem. Now, it seems to me that the proof of this theorem ...

**4**

votes

**1**answer

263 views

### When is Out$(SL_n(R))$ a torsion group ?

This question is a follow up question to this question. So my question is:
For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of ...

**12**

votes

**7**answers

1k views

### Superfluous definitions

It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...

**6**

votes

**3**answers

836 views

### Does “finitely presented” mean “always finitely presented”, considered in general

I'm wondering about the question, "If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" I know this is true for groups and for ...

**6**

votes

**0**answers

219 views

### Does there exist a commutative ring R such that SL_3(R) and SL_2(R) have the same finite subgroups?

This question is inspired, of course, by this question, and I don't know enough commutative algebra to know whether it's answered by silence dogood's answer to this follow-up question. If the answer ...

**3**

votes

**3**answers

1k views

### Why are divisible abelian groups important?

I just quote wikipedia:
"Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups."
I am asking for detail ...

**1**

vote

**2**answers

844 views

### Extension problem

As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...

**6**

votes

**2**answers

1k views

### Zero divisor conjecture and idempotent conjecture

Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$.
The wiki ...

**12**

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**0**answers

425 views

### A commutative monoid associated with a finite abelian group

Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
...

**2**

votes

**1**answer

782 views

### Automorphism theorem

Help me please to find reference for the proof of the following theorem:
Theorem. Let $\theta$ be a Leibniz cocycle on the Leibniz algebra L with values in $V,$ and assume $\theta^{\bot}\cap ...

**5**

votes

**2**answers

318 views

### dual of Z^I for uncountable I

Let $I$ be an infinite set. There is a homomorphism of abelian groups $\mathbb{Z}^{(I)} \to \hom(\mathbb{Z}^I,\mathbb{Z})$ which sends the basis element $e_i$ to the projection $p_i$. If $I$ is ...

**37**

votes

**7**answers

5k views

### Is there a slick proof of the classification of finitely generated abelian groups?

One the proofs that I've never felt very happy with is the classification of finitely generated abelian groups (which says an abelian group is basically uniquely the sum of cyclic groups of orders ...

**1**

vote

**2**answers

465 views

### Torsion-free and torsionless abelian groups

This question is motivated by my most spectacular answer on MO (:
Let $A$ be a module over $\mathbb Z$. $A$ is said to be torsion-free if $na=0$ implies $n=0$ or $a=0$ for any $n\in \mathbb Z, ...

**5**

votes

**3**answers

1k views

### Is a torsion free abelian group finitely generated, if all of its localizations at primes p are finitely generated over Zp?

Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map ...

**4**

votes

**1**answer

294 views

### Solvable subgroups of groups of polynomial automorphisms

Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?