Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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Number of n-th roots of elements in a finite group and higher Frobenius-Schur indicators
This is the second follow-up to this question on square roots of elements in symmetric groups and is concerned with generalisations to $n$-th roots. Let $G$ be a finite group and let $r_n(g)$ be the ...
- 13k
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Gluing hexagons to get a locally CAT(0) space
I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
The first two give the torus and the Klein bottle, respectively. What are the ...
- 10.1k
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Computing the Zariski closure of a subgroup of SL(n,Z)
Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...
- 3,201
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When does a homomorphism factor through a free group?
Let $f\colon\thinspace G\to H$ be a surjective homomorphism of finitely generated groups. Are there any methods to decide whether $f$ factors through a free group? That is, does there exist a free ...
- 35.9k
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Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
- 45k
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Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?
Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements
and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$.
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the ...
- 3,813
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Okounkov-Vershik approach to representation theory of $S_n$
This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
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Can a hyperbolic, one ended, one relator group, have a shorter trivial word?
Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the ...
user35370
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Almost commuting unitary matrices
Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
- 901
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Why does abelianization preserve finite products, really?
The abelianization functor $(-)^{ab} : \mathrm{Grp} \to \mathrm{Ab}$ is left adjoint to the inclusion of abelian groups into groups. As such, it preserves all colimits, but it doesn't generally ...
- 510
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Morphism from a surface group to a symmetric group, lifted to the braid group
Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
- 2,479
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Generating random finite groups
I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability $1/g(n)...
- 151k
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Amenability of groups
Let $G$ be non-amenable finitely generated group.
1) Is it true that there exists a sequence $S(n)$ of sets which generate $G$ and such that
$\frac{1}{|S(n)|}||\sum_{g\in S(n)} \lambda(g)||\...
- 4,705
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McKay conjecture for finite groups in the simplest case G=GL(2,F_p) ( puzzle: Borel knows about cuspidals)
The McKay conjecture and related (Alperin, Issacs-Navarro) are one of the "main problems in the representation theory of finite groups" (G.Navarro pdf).
Statement of the McKay conjecture is quite ...
- 24.9k
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Is there a big solvable subgroup in every finite group?
Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
- 11.3k
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If $S_\mathbb N$ is partitioned into finitely many pieces, must one piece contain a "skew copy" of every countable group?
Suppose $G$ and $H$ are groups. $C \subseteq H$ is called a skew copy of $G$ in $H$ if $C = hK$ for some $h \in H$ and some subgroup $K$ of $H$ with $K \cong G$.
Question 1: Suppose the infinite ...
- 18.5k
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Girth of the symmetric group
Let $n \in \mathbb N$ and $\{\sigma,\tau\} \subset {\rm Sym}(n)$ be a generating set.
Question: What is the maximal possible girth (if one varies $\sigma, \tau$) of the associated Cayley graph?
I ...
- 25.5k
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Is there an easy description of the structure of this infinite group?
Let $S_\infty$ denote the full symmetric group on countably many points and index the points by $\mathbb{N}$. For any weight (for lack of a better name) function $w:\mathbb{N}\rightarrow\mathbb R^+$, ...
- 5,191
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Are there enough additive permutations?
I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other ...
- 3,209
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Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series
Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm Sym}(\...
- 19.6k
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Are braid groups known to not be linear over $\mathbb{Z}$?
$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...
- 3,740
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Is there a "direct" proof of the Galois symmetry on centre of group algebra?
Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$
This induces a linear ...
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In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?
First let's recall some definitions. Let $G$ be a perfect group, so that
$$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$
for all abelian groups $A$ by universal coefficients. This means that when $A = ...
- 118k
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Density of first-order definable sets in a directed union of finite groups
This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i \...
- 2,200
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Finitely presented sub-groups of $\operatorname{GL}(n,C)$
Here are two questions about finitely generated and finitely presented groups (FP):
Is there an example of an FP group that does not admit a homomorphism to $\operatorname{GL}(n,C)$ with trivial ...
- 28.9k
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Understanding groups that are not linear
I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:
What are some interesting ...
- 397
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An easy proof that $S(n)$ does not embed into $A(n+1)$?
Rotman's book An Introduction to the Theory of Groups (Fourth Edition) asks, on page 22, Exercise 2.8, to show that $S(n)$ cannot be embedded in $A(n+1)$, where $S(n)$ = the symmetric group on $n$ ...
- 317
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Find a "natural" group that contains the quotient of the infinite symmetric group by the alternating subgroup
Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...
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Characterization of the transfer map in group theory
Let $i : H \to G$ be a subgroup of finite index. The transfer map is a special homomorphism $V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ ...
- 63.1k
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Small-index subgroups of SL(3,Z)
I would like to know the smallest-index subgroups of ${\rm SL}(3,\mathbb{Z})$.
The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$,
along the bottom row. I could not figure out ...
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Relationship between the cohomology of a group and the cohomology of its associated Lie algebra
Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is $\...
- 373
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What's so special about the forgetful functor from G-rep to Vect?
The following is some version of Tannaka-Krein theory, and is reasonably well-known:
Let $G$ be a group (in Set is all I care about for now), and $G\text{-Rep}$ the category of all $G$-modules (...
- 54.6k
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Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?
This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in ...
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Estimate for the order of the outer automorphism group of a finite simple group
It is known (given CFSG) that all non-abelian finite simple groups have small outer automorphism groups. However, it's quite tedious to list all the possibilities. Does anyone know a reference for a ...
- 4,728
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What did Frobenius prove about $M_{12}$?
I am interested in this paper which I can't read because it's in German:
Frobenius, G., Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02.
A free ...
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Isomorphism of $\mathbb{Z}\ltimes_A \mathbb{Z}^m$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^m$
here it's a question that I've posted in MSE but unfortunately got no answers:
Let $A$ and $B$ be matrices of finite order with integer coefficients.
Let $n\in\mathbb{N}$ and let $G_A=\mathbb{Z}\...
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Does the hypergraph of subgroups determine a group?
A hypergraph is a pair $H=(V,E)$ where $V\neq \emptyset$ is a set and $E\subseteq{\cal P}(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if ...
- 48.9k
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Categorical proof subgroups of free groups are free?
This is a crossport of this question from MSE.
Is there a categorical proof that subgroups of free groups are free?
How about the result that subgroups of free abelian groups are free abelian?
What ...
- 935
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Pair of short exact sequences of groups
Does there exist a pair of finite groups $G$ and $H$ satisfying both of the short exact sequences $1 \rightarrow G \rightarrow H \rightarrow A_4 \rightarrow 1$ and $1 \rightarrow G \rightarrow H \...
- 2,773
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Ultraproducts of finite cyclic groups
Let G be the (non-principal) ultraproduct of all finite cyclic groups of orders n!, n=1,2,3,... . Is there a homomorphism from G onto the infinite cyclic group?
user6976
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What invariants of a matrix or representation can be used to find its GL(n,Z)-conjugacy class?
First question: For a semisimple invertible $n \times n$ matrix with entries over a field K, its characteristic polynomial completely describes the similarity class of the matrix. For non-semisimple ...
- 7,320
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Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice
Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals.
More formally does ...
- 545
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Galois Bicentennial?
The 200-th anniversary of the birth of Galois will be on October 25th, 2011. For Abel's bicentennial birth year in 2002, Norway had a big conference and initiated the Abel prize. A cursory web search ...
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Finite groups $G$ so that $G$ has exactly two subgroups of a given order
Is there a finite group $G$ and a divisor $d$ of $|G|$ so that $G$ contains exactly two subgroups of order $d$?
The motivation for this question is an old qual problem (see http://www.math.wisc.edu/~...
- 20.4k
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Extending arithmetic functions to groups
Thinking along the lines of Tom Leinster's fascinating recent question, I'm wondering more generally about how to extend questions about natural numbers to groups, with the cyclic groups representing ...
- 6,792
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Half or more elements order two implies generalized dihedral?
The "generalized dihedral group" for an abelian group A is the semidirect product of A and a cyclic group of order two acting via the inverse map on A. A thus has index two in the whole group and all ...
- 7,320
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C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$
In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes M_2(\...
- 3,984
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Realizing braid group by homeomorphisms
Markovich and Saric proved the following remarkable theorem. Let $S$ be a compact surface of genus at least $2$ and let $MCG(S)=\pi_0(Homeo^{+}(S))$ be the mapping class group of $S$. There is then ...
- 203
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Proof of CFSG assuming every simple group is two-generated
It is well-known that one of the corollaries of the classification of finite simple groups (CFSG) is that every finite simple group can be generated by two elements. In a comment on an answer to an ...
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The finite groups with a zero entry in each column of its character table (except the first one)
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
