Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,823
questions
7
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1
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Groups acting on infinite dimensional CAT(0) cube complex
I have seen many examples where a finitely generated infinite group acts properly/freely by isometry on finite dimensional CAT(0) cube complexes. Examples of such groups are discussed in many articles....
- 329
1
vote
1
answer
93
views
Maximal abelian subgroups of an extraspecial group of order $2^{2m+1}$
I've found a proof of the structure of maximal abelian normal subgroups of an extraspecial group of order $2^{2m+1}$ in the book "Endlichen Gruppen I" by B. Huppert but there is a part of ...
1
vote
0
answers
161
views
Presentation of Chevalley groups over Bezout domains
Let $\Phi$ be a root system of type $A_1$, $A_2$, $B_2$ or $G_2$. For a (commutative, unital) ring $R$, consider the group $G_{\Phi}(R)$ defined by Steinberg's presentation as in [1, Theorem 12.1.1 ...
- 563
7
votes
1
answer
172
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Lifting SL2(k) to a subgroup of Witt vectors
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$Let $k$ be a finite field, and let $\W_n(k)$ be the degree $n$ Witt vectors over $k$ (so $\W_1(k) = k$).
Does there ...
- 35.8k
6
votes
3
answers
293
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Group such that factors in any product-decomposition are reducible
Motivation. Let us call a group $G = (G,\cdot)$ (product-)reducible if there are groups $H_1, H_2$, each having more than $1$ element, with $G \cong H_1\times H_2$. Otherwise, $G$ is said to be ...
- 44.5k
2
votes
2
answers
141
views
$String/CP^{\infty}=Spin$ or a correction to this quotient group relation
We know that there is a fiber sequence:
$$
... \to B^3 Z \to B String \to B Spin \to B^2 Z \to ...
$$
Is this fiber sequence induced from a short exact sequence?
If so, is that
$$
1 \to B^2 Z = B S^...
- 327
0
votes
0
answers
92
views
Order of elements in amalgamated free products
Reading the book "A Course in the Theory of Groups" by D. J. S. Robinson, I was looking at the proof of 6.4.3 (iii), which states (suppose we are in the case of two groups): if $G_1$ and $...
- 71
3
votes
2
answers
316
views
Are there $2^{\aleph_0}$ pairwise non-isomorphic countable groups containing every finite group?
Let us call a group $(G,\cdot)$ finitarily complete if $G$ is countable, and every finite group is isomorphic to a subgroup of $(G,\cdot)$.
Is there a collection of $2^{\aleph_0}$ pairwise non-...
- 44.5k
23
votes
2
answers
886
views
Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?
Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
- 1,115
10
votes
2
answers
723
views
Universal group such that every finite group is a quotient
We say that a permutation $\varphi:\mathbb{N}\to\mathbb{N}$ is finitary if there is $k\in\mathbb{N}$ such that $\varphi(i) = i$ for all $i\in\mathbb{N}$ with $i\geq k$. Let $I_\mathbb{N}$ denote the ...
- 44.5k
1
vote
1
answer
155
views
Suggestions about the set of all irreducible complex character degrees of a finite group
Let $G$ be a finite group, $\operatorname{cd}(G)$ be the set of all irreducible complex character degrees of $G$, and $\rho(G)$ be the set of all prime divisors of integers in $\operatorname{cd}(G)$. ...
- 577
1
vote
0
answers
109
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$p'$-automorphisms of pro-$p$ groups
Let $p$ be a prime and $G$ be a finitely generated pro-$p$ group admitting a continuous automorphism $\phi$ of finite order relatively prime to $p$. Let $\Phi(G)$ denote the Frattini subgroup of $G$. ...
- 499
8
votes
1
answer
443
views
Representation theory of $\mathrm{GL}_n(\mathbb{Z})$
I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
- 81
5
votes
2
answers
371
views
Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$
Let $k$ be a finite field. Do we always have $H^1(\operatorname{PSL}_2(k), k^3) = 0$, where $\operatorname{PSL}_2(k)$ acts on $k^3$ via the adjoint representation (= conjugation action on trace zero ...
- 35.8k
1
vote
0
answers
73
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Non-Noetherian closed subgroups of ${\rm GL}_{n}(\mathbb{F}_{q}[[T]])$
Let $\mathbb{F}_{q}$ be a finite field of order $q$, and $\mathbb{F}_{q}[[T]]$ be the ring of formal power series over $\mathbb{F}_{q}$. We say that a profinite group $G$ is Noetherian if any closed ...
- 499
1
vote
0
answers
129
views
Can a nontrivial abelian group have trivial Pontryagin dual? [closed]
Let $A$ be an abelian group, and suppose $$\mathrm{Hom}(A,\mathbb{Q}/\mathbb{Z})=0.$$
Does it follow that $A=0$?
This is true for $A$ finitely-generated, any subgroup of any product of copies of $\...
- 14.9k
6
votes
2
answers
224
views
Group homology for a metacyclic group
Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...
- 13.4k
7
votes
1
answer
179
views
Representations of the symmetric group with image in a given subgroup of $\operatorname{GL}_m$
Let $S_n$ be the symmetric group on $n$ elements. The irreducible representations of $S_n$ are parametrised by partitions $\lambda$ of $n$ and are defined already over the integers $\mathbb Z$.
Let $\...
- 291
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votes
1
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209
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For which finite groups $G$ is $M_n(\mathbb{Q}(\zeta))$ a factor of $\mathbb{Q}[G]$?
I am cross-posting this question from my MSE post here, in case someone here can answer it.
For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition:
$$
\mathbb{Q}...
- 103
5
votes
1
answer
176
views
Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic space
Let $\Gamma$ be a hyperbolic group. Let $g$, $\gamma\in \Gamma$ freely generate a non-abelian semigroup (in particular, they don't commute and have infinite order). Does the equation $g\gamma^n=h^m$ ...
- 215
5
votes
0
answers
186
views
What is known about the upper density of torsion elements in finitely generated groups?
Let $T\subset G$ be the set of all torsion elements in a finitely generated infinite group $G$, and let $B_n\subset G$ be the closed ball of radius $n$ around $1$ w.r.t. to the word metric for some ...
- 133
3
votes
0
answers
66
views
Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?
I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof).
Suppose $G$ is a ...
- 477
4
votes
1
answer
152
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 $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 7,836
2
votes
0
answers
133
views
Prime-to-$p$ quotients of ${\rm PSL}_{2}(\mathbb{Z}_{p})$
Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let ${\rm PSL}_{2}(\mathbb{Z}_{p})={\rm SL}_{2}(\mathbb{Z}_{p})/\{\pm 1\}$ be the projective special linear group over $\mathbb{Z}...
- 499
5
votes
1
answer
203
views
Function algebra of Furstenberg boundary $\partial_F \Gamma$: when is it a $W^*$-algebra?
Let $\Gamma$ be a non-amenable discrete group and consider its Furstenberg boundary $\partial_F \Gamma$. It is known that this is a compact topological space which is stonean (equivalently: extremely ...
- 433
9
votes
3
answers
260
views
Comparison between the operator norm and the $L^1$ norm on group algebras
Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question:
The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
- 1,146
2
votes
1
answer
292
views
Proving certain triangle groups are infinite
[Cross-posted from MSE]
Consider the Von Dyck group
$$ G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle $$
where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family of ...
- 4,327
9
votes
2
answers
549
views
Are these two methods for constructing Hadamard matrices known?
These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:
Context:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this ...
- 2,126
1
vote
0
answers
107
views
On the existence of non-arithmetic lattices in algebraic groups over $\mathbb{Q}$
$\newcommand{\Q}{\mathbb{Q}}\newcommand{\R}{\mathbb{R}}\DeclareMathOperator\PU{PU}$Let $G$ be a simple algebraic group over $\Q$ such that $G(\R) \simeq \prod_i G_i$, with each $G_i$ being the Lie ...
- 10.5k
10
votes
4
answers
810
views
Conjugation by elements of subgroups
Let $G$ be a group generated by a conjugacy class $C$. I am interested in studying this property:
for every $x,y\in C$ there exists $h\in \langle x,y\rangle$ such that $y=hxh^{-1}$.
Basically the ...
- 401
1
vote
0
answers
41
views
Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 535
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votes
1
answer
209
views
Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?
Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
- 243
2
votes
0
answers
76
views
Efficient decoding of the E8/Leech lattice
Background:
Our goal is to quantize a sequence of floating point numbers generated i.i.d. from a standard Gaussian source and minimize the MSE reconstruction error. We can use two bits for each sample....
- 121
0
votes
0
answers
85
views
Invariants of primary groups
In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...
- 31
1
vote
1
answer
117
views
Equivalence of dihedral and symmetric group actions on a specialized real algebra
Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help.
consider first the case where the digit 7 is not allowed, simplifying the ...
- 153
6
votes
1
answer
313
views
Classifying abelian (but non-central) group extensions using homotopy theory
Let $G$ be a group and let $A$ be an abelian group equipped with an action of $G$. Group extensions
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
inducing the ...
- 42.8k
1
vote
0
answers
72
views
cycle types of all words in a permutation group
I have been working with permutation groups. For a given $G\subset S_n$, what I have been computing depends only on the conjugacy class of $G$.
Say all permutation groups in this question are ...
- 2,145
3
votes
1
answer
116
views
Topological amenability of actions - forgetting topology
Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
- 516
9
votes
1
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406
views
Growth of powers of symmetric subsets in a finite group
(This question was originally asked on Math.SE, where it was answered in the abelian case)
Let $G$ be a finite group, and let $A$ be a symmetric subset of $G$ containing the identity (i.e., $A^{-1}=A$ ...
- 2,474
1
vote
0
answers
118
views
Exotic automorphisms of an extension of Thompson's group $V$
Recall that R. Thompson's group $V$ acts transitively on the set $\mathbb{Q}_2$ of dyadic rationals contained in the unit interval $[0,1)$.
Main question. Does there exist a non-trivial $V$-...
5
votes
0
answers
134
views
Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
2
votes
0
answers
65
views
Euler class of extension of free nilpotent groups
Fix some $n \geq 2$. For $k \geq 1$, let $N_k$ be the free $k$-step nilpotent group on $n$ generators, i.e., the quotient of the free group $F_n$ by the $(k+1)^{\text{st}}$ term $\gamma_{k+1}(F_n)$ ...
- 21
5
votes
2
answers
456
views
Non-trivial extension of representations have same central character
Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user515481
4
votes
0
answers
164
views
Square hidden number problem
Suppose I have a mystery number $m$ modulo $p$ that I wish to find. I know the value of $m+x_i^2$ where $x_i$ is randomly chosen modulo $p$ for some large number of different $x_i$, $N$ many, $N \gg \...
- 581
1
vote
0
answers
67
views
Bias of $a^k / q$ modulo $q$?
Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$...
- 581
2
votes
3
answers
288
views
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$
of cardinality $2m\ge 6$ where $m$ is odd.
Question 1. Is it true that $G$ always has a subgroup $H$ of index 2
...
- 13.4k
3
votes
1
answer
228
views
Does a quasi-split reductive group scheme admit a maximal torus?
Let $G \to S$ a reductive group scheme over arbitrary base. Following the conventions from Conrad's Reductive Group Schemes notes, we define a Borel subgroup to be an $S$-subgroup scheme $B \subseteq ...
- 535
0
votes
0
answers
83
views
The relation between two characteristic subgroups in finite p-group
Suppose $G$ is a finite $p$-group. Let
\begin{align*}
\mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle.
\end{align*}
There are examples such that $|G|\leq |\...
- 71
2
votes
1
answer
259
views
Does any finite group of order $2m$ with odd $m$ have a subgroup of index 2? [closed]
Let $G$ be a finite group of order $2m$ where $m>1$ is an odd natural number.
Question. Is it true that any such $G$ has a subgroup $H$ of index 2?
If yes, I would be grateful for a reference or ...
- 13.4k
3
votes
1
answer
175
views
What is the minimum possible k-rank of a quasi-split reductive group over a field?
It is not possible for a quasi-split reductive group $G$ over a field $k$ to be anisotropic (unless it is solvable, hence its connected component is a torus). Indeed, there exists a proper $k$-...
- 535