Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,042
questions
13
votes
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answer
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Number of trivializations of a trivial word in the free group
Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...
- 17.3k
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votes
4
answers
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Elements of infinite order in a profinite group
Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...
- 10.9k
12
votes
1
answer
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are the congruence subgroups $\Gamma(n)$ characteristic inside $\mathrm{SL}_2(\mathbb{Z})$?
For which $n$ is the "principal congruence subgroup" $\Gamma(n)\le \mathrm{SL}_2(\mathbb{Z})$, the subgroup consisting of matrices congruent to the identity modulo $n$, characteristic? I.e., for ...
- 6,567
12
votes
2
answers
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Existence of finite index torsion-free subgroups of hyperbolic groups
Question. Is it true that each infinite hyperbolic group
has a torsion-free subgroup of finite index?
Are there counterexamples, or positive results for some large subclasses of hyperbolic groups?
For ...
- 28.6k
12
votes
0
answers
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views
Non split extension isomorphic (as a group) to a split extension
$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...
- 1,176
12
votes
1
answer
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How many generators does a direct product of alternating groups need?
P. Hall gave a formula for the number of generators of $G^n$ for any finite simple group $G$. One famous example is the fact that $A_5^{19}$ is 2-generated, but $A_5^{20}$ is not. The question of ...
12
votes
4
answers
3k
views
When does Pontryagin duality generalize?
Let $T$ be a locally compact abelian (LCA) group. For any other LCA group $G$, let
$\hom(G,T)$ be the set of continuous homomorphisms $G\to T$. With the compact-open
topology, $\hom(G,T)$ is ...
- 5,569
12
votes
1
answer
563
views
Is residual finiteness a quasi isometry invariant for f.g. groups?
A "residually finite group" is group for which the intersection of all finite index subgroups is trivial. Suppose $G$ and $G'$ are two quasi-isometric finitely generated groups. Does the residual ...
- 4,454
12
votes
0
answers
779
views
Commutator subgroup of a surface group
Let $\Sigma_{g,n}$ denote a compact orientable genus $g$ surface with $n$ boundary components. Assume that $g \geq 1$ and fix a basepoint $p \in \Sigma_{g,n}$. Define $S \subset [\pi_1(\Sigma_{g,n},...
- 42.8k
11
votes
1
answer
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Classification of (not necessarily connected) compact Lie groups
I am looking for a classification of compact (not necessarily connected) Lie groups. Clearly, all such groups are extensions of a finite "component group" $\pi_0(G)$ by a compact connected ...
- 473
11
votes
2
answers
611
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Quasinilpotent elements of group C-star algebras
If $G$ is a discrete torsion-free group, can its (reduced or full) group C-star algebra contain non-zero quasinilpotent elements? I've seen various examples in the group von Neumann algebra setting (...
- 25.3k
11
votes
2
answers
726
views
Knot groups with big number of generators
I start by saying that I am not an expert in this field and I apologize if the question is too elementary.
Let $K$ be a knot in $S^3$. I denote by $\pi_1(K)$ the knot group, which is the fundamental ...
- 674
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votes
5
answers
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What are the normal subgroups of a direct product?
Let $N$ be a normal subgroup of $G \times H$, and let $\pi_1: G \times H \to G$ and $\pi_2: G \times H \to H$ be the canonical projections. Then $\pi_1(N)$ is normal in $G$ and $\pi_2(N)$ is normal in ...
- 1,851
11
votes
3
answers
1k
views
A problem on a specific integer partition
Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that :
$d_{i}\vert n$
$1=d_{1}<d_{2} \le d_{3} \le ... \le d_{r}$...
- 25.5k
11
votes
5
answers
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Structure of the adjoint representation of a (finite) group (Hopf algebra) ?
Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation.
Question 1: what is known about this representation ...
- 22.9k
11
votes
4
answers
2k
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Are measurable automorphism of a locally compact group topological automorphisms?
Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \to G$, which is measurable and has an inverse, which ...
- 11k
10
votes
1
answer
504
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Counting symmetric subgroups of symmetric groups
This question is related to, but much more specific than, this one.
For $k \leq n$, let $a(k,n)$ denote the number of conjugacy classes of subgroups of the symmetric group $S_n$ which are isomorphic ...
- 2,693
10
votes
1
answer
547
views
Extensions isomorphic as groups but not congruent or pseudo-congruent
I'm looking for an example of a finite abelian group A and a finite group G acting trivially on A such that there are two extensions $E_1$ and $E_2$ with base A and quotient G (i.e., they are both ...
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votes
3
answers
536
views
Is each finite group multifactorizable?
Definition. A finite group $G$ is called multifactorizable if for any positive integer numbers $a_1,\dots,a_n$ with $a_1\cdots a_n=|G|$ there are subsets $A_1,\dots,A_n\subset G$ such that $A_1\cdots ...
- 40.2k
10
votes
1
answer
951
views
Acyclic Finite Groups
A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also ...
- 2,590
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votes
2
answers
701
views
Is there a non-degenerate quadratic form on every finite abelian group?
Let $G$ be a finite abelian group. A quadratic form on $G$ is a map $q: G \to \mathbb{C}^*$ such that $q(g) = q(g^{-1})$ and the symmetric function $b(g,h):= \frac{q(gh)}{q(g)q(h)}$ is a bicharacter, ...
- 25.5k
10
votes
2
answers
696
views
Can any finite lattice be realized as an intermediate subgroups lattice?
Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...
- 25.5k
10
votes
3
answers
1k
views
subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,145
10
votes
4
answers
2k
views
residually finite-by-$\mathbb{Z}$ groups are residually finite
I believe I read somewhere that residually finite-by-$\mathbb{Z}$ groups are residually finite. That is, if $N$ is residually finite with $G/N\cong \mathbb{Z}$ then $G$ is residually finite.
However, ...
- 2,742
9
votes
1
answer
572
views
Is $\operatorname{PSL}(2,q)$ the most quasirandom group?
Is the following statement true?
Every finite group $G$ has a non-trivial irreducible representation of dimension $O(\lvert G\rvert^{1/3})$.
Context: Groups with no small irreducible representations ...
- 946
9
votes
0
answers
365
views
Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
- 64.8k
9
votes
7
answers
3k
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Hopfian and Co-Hopfian groups (examples)
Hi,
I'm looking for examples of groups that are both Hopfian and Co-Hopfian. I have a non trivial (and beautiful, at least to me) example: $\mathrm{SL}(n,\mathbb{Z})$ (with $n>2$).
Do you know ...
9
votes
2
answers
2k
views
Chevalley Groups over an arbitrary ring.
My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the ...
- 2,458
9
votes
2
answers
852
views
transfer kernels and the Schur multiplier
Let $\Gamma$ be a finite $2$-group, and let $G$ be any subgroup
of index $2$. Moreover, let Ver$: \Gamma/\Gamma' \to G/G'$
denote the group theoretical transfer, and let $M(\Gamma)$ be
the Schur ...
9
votes
1
answer
402
views
Left orderable linear groups
Are all torsion-free finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsion-free congruence subgroups of $SL_n(\mathbb{Z})$ left orderable?
user6976
9
votes
2
answers
618
views
Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
- 52.1k
9
votes
3
answers
2k
views
Character tables and simple groups.
Is it known if there are two finite non-isomorpic non-abelian simple groups with the same character table? Does this answer change if the subsidiary information (like the orders and sizes of the ...
- 91
9
votes
2
answers
707
views
Return probabilities for random walks on infinite Schreier graphs
Question: Is there a sequence $(\delta_n)_n$ of real numbers with $\delta_n \to 0$ as $n \to \infty$, such that the following holds:
Let $F$ be a free group on two generators, let $F \curvearrowright ...
- 25.2k
9
votes
0
answers
393
views
'Almost-isomorphic' groups
What can be said about pairs of non-isomorphic groups which are epimorphic images of one another and which also embed into one another?
Can such pairs of groups be 'classified' in some sufficiently ...
- 19.3k
9
votes
2
answers
3k
views
The definition of a group object is wrong?
An old MO answer by Noah Snyder makes a claim I don't completely understand, but mostly because I don't know any examples. The answer claims that in some examples of (things that one would want to ...
- 114k
9
votes
2
answers
517
views
A "subtle" isomorphism testing problem: $\mathbb{Z}\ltimes_{A} \mathbb{Z}^5\cong \mathbb{Z}\ltimes_{B}\mathbb{Z}^5$ or not?
EDIT: I've made a mistake with the matrices. Now it is corrected.
A couple of days ago I asked this question. There, answerers gave me excellent hints to solve that case and others too. But I've found ...
8
votes
3
answers
505
views
Are two elements of a group determined up to simultaneous conjugacy by the conjugacy classes of all of their products?
Let $G$ be a group (if it helps, assume that $G$ is a Lie group or finite). Is a pair of elements $(g, h) \in G \times G$ determined up to simultaneous conjugacy by the conjugacy class of every ...
- 114k
8
votes
1
answer
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views
A question on "$p$-group bounds"?
Suppose $n_p(G)$ is the number of elements of order $p$ in a group $G$.
Does there for every prime $p$ exist a $\epsilon_p > 0$, such that for any group $G$ $n_p(G) > (1 - \epsilon_p(G))|G|$ ...
- 2,618
8
votes
2
answers
309
views
Cubic almost-vertex-transitive graphs with given spanning tree
Consider the infinite 3-regular tree. Pick a vertex $C$, the "center".
For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced ...
- 21.4k
8
votes
2
answers
612
views
Ends of finitely generated torsion groups
It is known that the number of ends of a finitely generated group is 0,1, 2 or $\infty$.
Problem 1. What is known about the number of ends of infinite finitely generated torsion groups?
In ...
- 40.2k
8
votes
2
answers
558
views
Is residual finiteness a property of "many" finitely presented groups?
Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold:
$G$ is ...
- 11.2k
8
votes
1
answer
199
views
Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple?
Does the (simply connected compact) Lie group $E_7$ contain a finite subgroup $G \subset E_7$ such that the $56$-dimensional irrep of $E_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-...
- 52.2k
8
votes
4
answers
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Are all group monomorphisms regular, constructively?
Are all group monomorphisms regular, constructively?
By "constructive" I mean something that would go through in CZF for example.
[added Oct 6]
A sketch of a standard proof (such as referenced in ...
- 91
8
votes
1
answer
245
views
Algorithmically handling the Spin groups in larg(ish) dimensions
Question: Is there a reasonably efficient algorithmic representation of $\mathit{Spin}_n$? By this I mean, a way to store its elements and operate on them (multiply, inverse, maybe compute ...
- 28.7k
8
votes
1
answer
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Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
$$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \...
- 1,269
7
votes
2
answers
492
views
A linearly orderable monoid which does not embed into a linearly orderable group
It is known (after an example of A.I. Mal'cev) that there exist cancellative semigroups which do not embed into a group. On the other hand, it is not difficult to see that every linearly orderable ...
- 9,446
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votes
3
answers
1k
views
Is there a formula for the size of Symplectic group defined over a ring $Z/p^k Z$?
Is there a formula for the size of Symplectic group defined over a finite ring $Z/p^k Z$?
- 71
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votes
2
answers
859
views
Which 3-regular graphs are Schreier coset graphs?
Given a group $G$ and a subgroup $H$ the Schreier coset graph (w.r.t. some set $S$ of $G$) is the directed (and labelled) graph whose vertices are the cosets of $H$ (i.e. the set $G/H$) and $x \sim y$ ...
- 4,342
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votes
2
answers
437
views
Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring
Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" ...
- 1,593
7
votes
0
answers
464
views
Abstract characterization of group von Neumann algebra (II1 factor)
The group von Neumann algebra $L\Gamma$ is a factor if and only if the group $\Gamma$ is ICC (i.e. infinite conjugacy class property). Moreover if $\Gamma$ is nontrivial then $L\Gamma$ is a $\mathrm{...
- 25.5k