All Questions
Tagged with group-theory or gr.group-theory
8,181 questions
6
votes
1
answer
200
views
Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?
This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?
- 103
5
votes
2
answers
423
views
Do all Brauer relations for finite groups vanish under the augmentation map?
The Burnside ring $B(G)$ of a finite group $G$ is the ring of all finite $G$-sets under disjoint union and Cartesian product. It is well known that $\mathbb{Q} \otimes B(G)$ has as basis $\{[G/H] \mid ...
- 161
8
votes
2
answers
1k
views
Schreier-Sims algorithm for solving Rubik's cube
Currently I am studying the Schreier-Sims algorithm. To gain a deeper understanding, I am trying to look into applications of this algorithm, among which its use for solving the Rubik's Cube is ...
- 83
2
votes
0
answers
101
views
What is the natural module?
Lemma 2.9 of [1]:
Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...
- 217
8
votes
1
answer
310
views
Decomposing the homology of a finite-index subgroup into isotypic components
$\newcommand\C{\mathbb{C}}$Let $\Gamma$ be a discrete group and let $M$ be a $\C[\Gamma]$-module. Let $G \lhd \Gamma$ be a finite-index normal subgroup with quotient $Q = \Gamma/G$. The conjugation ...
- 83
3
votes
1
answer
423
views
Groups with no homomorphisms onto $\mathbb{Z}/p\mathbb{Z}$
Does there exist any term for finite groups with no non-trivial homomorphisms into $\mathbb{Z}/p\mathbb{Z}$ for a fixed prime $p$, or any term related to this property (so that I could write "...
- 16.9k
3
votes
1
answer
177
views
Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
- 254
3
votes
0
answers
88
views
Homological criterion for finite generation
Let $G$ be a (discrete) group. Assume that for all finitely generated $\mathbb{Z}[G]$-modules $M$, the homology group $H_1(G;M)$ is finitely generated. Does it follow that $G$ is finitely generated?
...
- 31
4
votes
0
answers
83
views
Additive characters from coarse quotient maps
Let's consider a (finitely generated) group $\Gamma$ and a
coarse quotient map
$q\colon\Gamma\to\mathbb{R}$.
I'm interested in the 1-cocycle
$\sigma\colon\Gamma\to\ell_\infty\Gamma$,
defined by $\...
- 10.1k
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
7
votes
1
answer
224
views
Generating set of permutation group such that generators do not "contain" other group elements
Let $(G, X)$ be a permutation group with domain $X$. Let $O=\{o_1,\dots,o_m\}$ be the set of orbits of $G$. I am interested in generating sets $S$ with the following property:
Let $g\in S$ be a ...
- 5,792
2
votes
1
answer
273
views
Equivariant Smith normal form?
Let $F$ be a finitely generated free $\mathbb{Z}$-module on which the group $G$ of two elements acts via group homomorphisms. Let $F'$ be a $G$-invariant submodule. By Smith normal form we know that ...
- 3,031
6
votes
1
answer
146
views
If $X$ is a hyperbolic, locally finite graph with $\partial X \cong S^1$, and $G$ acts cocompactly but not properly on $X$, what can we say?
It is an important and deep fact of geometric group theory that if the Gromov boundary of a hyperbolic group $G$ is a circle, then $G$ is virtually Fuchsian [Tukia, Gabai, Casson-Jungreis...]. I am ...
- 609
2
votes
0
answers
100
views
Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
- 21
4
votes
1
answer
523
views
Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus？ [closed]
Is automorphism on a compact group necessarily homeomorphism? I don't think so，but I think it is possible on the N-dimensional torus.
- 83
5
votes
0
answers
140
views
Classification of visible actions for *reducible* representations?
Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
- 3,230
13
votes
4
answers
843
views
What is a "general" relation algebra?
I'm trying to understand why (or if) the axioms of relation algebras are "the right ones." For example, we can back up the idea that the group axioms exactly capture the notion of "...
- 21.1k
7
votes
2
answers
200
views
When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?
Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space.
Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...
- 431
8
votes
1
answer
217
views
Bilinear forms on abelian $p$-groups: Images of weak metabolizers in the Frattini quotient
Suppose $G$ is a finite abelian $p$-group for $p$ an odd prime and $b : G \times G \to \mathbf{Q}/\mathbf{Z}$ is a non-degenerate symmetric bilinear form. A subgroup $H \leq G$ is a weak metabolizer ...
- 365
11
votes
1
answer
248
views
Recognising the elements of the Grigorchuk group
The Grigorchuk group $\mathfrak{G}= \langle a,b,c,d \rangle$ is a group of automorphisms of the infinite rooted binary tree $\mathcal{T}_2$. Every element of $\mathfrak{G}$ can be represented by a ...
- 8,401
4
votes
1
answer
289
views
Is every finite simple group contained in a group of the form $\operatorname{PSL}(n,p)$?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\PSL{PSL}$Is every finite simple group contained in a group of the form $\PSL(n,p)$ for some integer $n\ge 1$ and prime $p$?
More generally, I'd like ...
- 7,527
1
vote
0
answers
103
views
Normality of the intersection between a Carter subgroup and the nilpotent residual of a solvable group G
In his book "Group Theory," Schenkman, in the proof of Theorem VII.4.a, states that in a finite solvable group $G$, the intersection of a Carter subgroup $C$ (i.e., a self-normalizing and ...
8
votes
1
answer
353
views
Structure of a single automorphism of a finite abelian p-group
A finite abelian $p$-group $H$ is homogenous when it is the direct sum of cyclic groups of the same order $p^r$, i.e. $H \cong \big(\mathbb{Z}/p^{r}\mathbb{Z}\big)^{e}$. Every finite abelian $p$-group ...
- 365
1
vote
1
answer
221
views
Density of numbers with a large prime factor in specified arithmetic progression
I am looking for an answer to the following question.
Fix some coprime integers $a$ and $b$ and let $S_x$ be the set of positive integers $n<x$ such that there exists a prime factor $p$ of $n$ with ...
4
votes
1
answer
216
views
"Universal" abelian p-groups
Let $p$ be a prime number. I am interested in the abelian groups $G$ with the following property:
(U) every finite abelian $p$-group $A$ admits a monomorphism $A\hookrightarrow G$.
In other words, ...
- 1,294
1
vote
0
answers
92
views
The existence of such homomorphism [closed]
Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of non-trivial homomrphism f:G->H were H is some nontrivial ...
- 61
2
votes
0
answers
115
views
Test words in free profinite groups
Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $...
- 355
2
votes
0
answers
118
views
What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
0
votes
1
answer
161
views
How to determine if a set is a sumset
Let $G$ be a commutative group (assume whatever you want on $G$ if needed. I am mainly interested in $G = \mathbb{Z}/n\mathbb{Z}$).
Let $k$ be a fixed integer.
Let $(a_1, \dots, a_{k^2})$ be a list of ...
- 527
4
votes
1
answer
593
views
Commutativity of the wreath product
(Cross-posted from MSE, with isomorphism replaced by conjugate: https://math.stackexchange.com/questions/4928697/commutativity-of-the-wreath-product?noredirect=1#comment10531931_4928697 )
Let $G$ be a ...
- 891
12
votes
2
answers
621
views
Group with a translation invariant ultrafilter
Let $G$ be an infinite, discrete, countable group. Can $G$ have a translation-invariant ultrafilter? An ultrafilter $\mathcal{F} \subset 2^G$ is translation-invariant if $A \in \mathcal{F}$ implies $g ...
- 1,322
0
votes
1
answer
379
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
2
votes
0
answers
153
views
How good is approximation of distance function on the Cayley graph by "Fourier" basis coming from the irreducible representations?
Consider finite group $G$ , symmetric set of its elements $S$, construct a Cayley graph.
Consider $d(g)$ - word metric or distance on the Cayley graph from identity to $g$.
As any function on a group ...
- 24.8k
-1
votes
1
answer
146
views
Learning Schreier–Sims Algorithm [closed]
I am trying to learn Schreier–Sims Algorithm, and looking at Mathematics and Such - Schreier–Sims algorithm.
In the 3rd line of red/yellow box in 2nd figure in the link:
I am not sure why the 2nd and ...
- 83
2
votes
1
answer
111
views
Structure of elements of a finite group not contained in any conjugate of a proper subgroup
Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$,
$$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$
is properly ...
1
vote
1
answer
114
views
A correspondence between projective representations of $G$ with those of its universal cover
Let $G$ be a connected Lie group and $\mathcal{H}$ be a Hilbert space. Let $U(\mathcal{H})$ denote the the group of all unitary operators on $\mathcal{H}$ with function composition (i.e., $\hat{U}:\...
- 287
-2
votes
1
answer
241
views
Does a group representation being transitive on a basis imply irreducibility?
Let $G$ be an infinite discrete group and $\pi$ a representation of $G$ on the Hilbert space $H$.
Suppose that the group representation is transitive on an orthonormal basis $B = \{e_j\}_{j=1}^{\infty}...
5
votes
2
answers
574
views
Elements that are automorphic images
Consider an finite abelian group $G$ and two elements $x,y \in G$. Is there a way to check whether there exists a $\phi \in \mathrm{Aut}(G)$ such that $\phi(x) = y$?
Here are some necessary conditions ...
- 51
3
votes
1
answer
182
views
Schur cover of alternating groups
Wilson's book "The finite simple groups" gives (in section 2.7) a description of the double cover of the alternating groups. First, one constructs a double cover $2S_n$ of the symmetric ...
- 7,527
14
votes
2
answers
851
views
Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
- 143
3
votes
1
answer
197
views
Is the exponential map of a locally compact group a local homeomorphism?
We consider a locally compact abelian group $G$. We equip the real vector space $A(G)$ of continuous group homomorphisms $\mathbb{R}\to G$ with the topology of uniform convergence on compact subsets ...
- 3,031
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
7
votes
2
answers
353
views
Finite normal subgroup of mapping class group
Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\...
- 605
7
votes
0
answers
145
views
Transitive groups with fixed-point free elements of prime power order
A well-known result of Fein, Kantor and Schacher says that if $G$ is a finite group which acts transitively on a set $X$, then $G$ contains an element of prime power order without fixed letters. ...
- 4,547
6
votes
0
answers
176
views
Normality and small doubling
Suppose that $A$ is a finite, generating subset of a group $G$, and that $H$ is a subgroup such that $A^2$ is a union of left $H$-cosets; moreover, $H$ is maximal subject to this property. Is it true ...
- 23k
9
votes
1
answer
335
views
Finite p-group $G$ of exponent $>p$ with all elements outside $\Phi(G)$ of order $p$
Does there exist a finite $p$-group $G$ of exponent $>p$, such that $o(g)=p$ for all $g\in G\setminus\Phi(G)$?
- 171
7
votes
1
answer
445
views
What is known/expected on the co-growth series of the braid group?
The co-growth series of finitely generated group with respect to generating set $S$ is generating function for the number of words of length $n$ which are equal to 1 in the group.
Its studies ...
- 24.8k
9
votes
2
answers
438
views
Irreducible tensor product representations in finite simple groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...
15
votes
1
answer
853
views
Algebraic structure on conjugacy classes
informally speaking, what algebraic structure does the set of conjugacy classes of a group carry?
Formally, I'm interested in natural operations on conjugacy classes. Let $\mathsf{Grp}$ be the ...
- 6,406
0
votes
0
answers
68
views
A reference for this statement (representations of universal central extensions)
Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact:
"Every projective unitary ...
- 287