All Questions
Tagged with ra.rings-and-algebras gr.group-theory
309 questions
-3
votes
0
answers
139
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
2
votes
1
answer
153
views
On generation of $A_n$ by elements of prime order
There is a question regarding generation of finite simple groups with elements of prime order. Recently, Guralnick, Shareshian, Woodroofe and Teräväinen made advances in this direction. We have, for ...
- 329
8
votes
1
answer
321
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
1
vote
0
answers
89
views
The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
3
votes
0
answers
161
views
Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
3
votes
2
answers
468
views
How fast does the number of "fixed" points grow compared to the size of the ball in the following group?
I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...
- 823
0
votes
1
answer
380
views
Finitely generated and finitely presented [closed]
For a group, it seems fairly clear that finitely presented implies finitely generated.
But what about the converse? Is there a finitely generated group that is not finitely presented.
(Let's say ...
- 365
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
3
votes
1
answer
426
views
Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...
- 985
1
vote
1
answer
115
views
Equivalent definition of Spin group in terms of automorphisms
Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...
- 1,903
4
votes
1
answer
163
views
Examples of Noetherian integral group ring
I need to study the integral group ring of the fundamental group of a manifold. My knowledge of group and ring theory is very limited. I am looking for some examples of groups $G$ for which $\Bbb ZG$ ...
- 1,097
2
votes
0
answers
101
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
- 667
4
votes
0
answers
453
views
Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
4
votes
1
answer
373
views
How to have MAGMA work with subgroup of ATLASGroups?
I'm trying to work with various maximal subgroups of the Thompson sporadic group. The command Group("Th"); which works for some of the sporadic groups, ...
- 143
7
votes
1
answer
2k
views
If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?
Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...
- 97
4
votes
2
answers
412
views
Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\...
- 985
4
votes
1
answer
211
views
Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 10.6k
2
votes
0
answers
137
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,485
2
votes
0
answers
60
views
upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups
Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
- 823
2
votes
0
answers
156
views
The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$
Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...
- 823
10
votes
2
answers
869
views
When are two semidirect products of two cyclic groups isomorphic
(I have posted this question in Math Stack Exchange, only to have received no answer.)
It is well known that a semidirect product of two cyclic groups $C_m$ and $C_n$ has the form
$$
C_m \rtimes_k C_n ...
- 525
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
7
votes
1
answer
633
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
2
votes
1
answer
244
views
Markov property for groups?
My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
- 309
2
votes
0
answers
308
views
A question on Giles Gardam counter example to the Unit conjecture of Kaplansky
The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an ...
- 356
8
votes
2
answers
645
views
Analogous results in geometric group theory and Riemannian geometry?
As you can see from my other question I concern mmyself with the following article at the moment:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–...
- 309
9
votes
1
answer
735
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 6,741
3
votes
1
answer
353
views
Question to limit groups (over free groups)
My question refers to the following article (to page 26: proof of Theorem 4.1):
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10....
- 309
5
votes
1
answer
298
views
Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence?
I have a question that is related to the topic of limit groups:
Let $G$ and $H$ be finitely generated groups and let $(\varphi_n: G \to H)_{n \in \mathbb{N}}$ be a sequence of group epimorphisms. Does ...
- 309
2
votes
0
answers
144
views
Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$
Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.)
(Please choose any irrep ...
- 21
0
votes
0
answers
116
views
Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 10.5k
1
vote
0
answers
112
views
Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
- 356
0
votes
0
answers
96
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 356
1
vote
1
answer
252
views
Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 823
16
votes
1
answer
850
views
A "simpler" description of the automorphism group of the lamplighter group
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references.
The lamplighter group is defined by the ...
- 823
11
votes
1
answer
159
views
Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?
A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that
$$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$
An example of such an object is ...
- 301
3
votes
1
answer
148
views
Commensurator of $\mathrm{SL}_2(\mathbb{Z})$ on $\mathrm{GL}_2^+(\mathbb{Q})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$I am trying to find the commensurator of $\SL_{2}(\mathbb{Z})$ on $\GL_{2}^+(\mathbb{R})$.
So far I have been able to prove that $\GL_{2}^+(\...
-1
votes
1
answer
199
views
Isomorphism between subgroups by preserving index
Let $C$ be a subgroup of a group $A$ such that $C\cong A/\{\pm1\}$ and $D$ be a subgroup of $B$ such that $D\cong B/\{\pm1\}$. Let $\pi:A\to B$ be a group homomorphism, and let $\pi$ induce a ...
- 309
6
votes
1
answer
447
views
Calculating the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$
I want to calculate the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$ when its generators $G_i$ (whose number is finite) are known.
For any matrix $S$ that commutes with the group: $G_iS$ =...
- 330
5
votes
2
answers
274
views
Glauberman-Thompson normal $p$-complement theorem for $p=2$
I asked this question on Math StackExchange yesterday. As suggested by Professor Derek Holt, this question may be more suitable for this site. So I ask this question here again, but more details and ...
- 151
7
votes
1
answer
404
views
Do rational group algebras have an outer automorphism?
In the article "Automorphism groups of simple algebras and group algebras" (1978), Janusz conjectures the following:
The group algebra $\mathbb{Q} G$ for a non-trivial finite group has an ...
- 26.5k
6
votes
0
answers
185
views
How much information can we extract about such a group?
Let $G$ be a group with the property : Given any positive integers $m, n$ and $r$ there exists elements $g, h\in G$ such that $|g|=m, |h|=n$ and $|gh|=r$
Where $|g|$ is the order of the group $\langle ...
- 307
0
votes
1
answer
184
views
Hopf algebra of representative k-valued functions of an abstract group
Let $G$ be an abstract group. If we can embed $G$ into a group $H$, in a way that we had $G$ and $H$ of the same Hopf algebra of representative $k$-valued functions ($R(G)\sim R(H)$ as Hopf algebras). ...
- 29
0
votes
2
answers
155
views
Examples of isomorphic non-equivalent twisted group algebras
Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
- 253
2
votes
1
answer
161
views
When are elements of a (perfect) semidirect product simple commutators?
I am migrating this question from math stackexchange...
I have a semidirect product and I have shown that it is perfect. However, I would like to know whether every element is a simple commutator (...
- 23
3
votes
1
answer
187
views
Profinite completion of Baumslag-Solitar group as a profinite HNN-extension
I apologize if this question is basic in some sense. I was looking for an example of a non-proper HNN-extension and I found this.
In the comments, markvs mentioned the Baumslag-Solitar group $B(2,3)$. ...
- 329
1
vote
0
answers
78
views
tensor dimension/reshaping group
Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
- 11