# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

Questions about the branch of algebra that deals with groups.

7,326
questions

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What is the Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$? The techniques used here give $K_2(\mathbb{Q})$ as a lower bound, but it’s probably bigger than that, especially since the universal cover ...

-4
votes

0
answers

56
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Recently, I am try to solve a problem in character theory:
Character extension about $Q_8$
In this problem we have that $G=G/G'\cap N\lesssim G/G'\times G/N=G/G'\times Q_8$. If $G=N\times Q_8$, then $\...

2
votes

0
answers

85
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In the proof of Lemma 2.1 in
Ershov, Mikhail; He, Sue, On finiteness properties of the Johnson filtrations, ZBL06904638,
the authors claim the following (without proof).
Let $G$ be a finitely ...

2
votes

0
answers

65
views

Let $H(\mathbb{Z})$ be the discrete Heisenberg group. What are the finite dimensional irreducible unitary representations of $H(\mathbb{Z})$? Do they all arise from the coordinate-wise quotient map to ...

4
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1
answer

312
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$\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A_4$ ...

3
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1
answer

108
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In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...

1
vote

0
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93
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$G$ is a finite solvable group. Let $\{P_{1}, P_{2}, \dotsc , P_{s}\}$ be a Sylow basis of $G$. We have that $G=P_{1}P_{2}\dotsm P_{s}$. Set
\begin{equation}
\begin{aligned}
%% The alignment is ...

2
votes

0
answers

68
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Do the Suzuki and Ree groups of Lie type have associated Lie algebras over finite fields in the same way that the other groups of Lie type do? These algebras would be 5-dimensional over $\mathbb{F}_{2^...

8
votes

2
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148
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For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...

10
votes

3
answers

419
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$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,
$$\...

1
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0
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65
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I started reading this paper by Andrea Lucchini. Title: Generators for Finite Groups with a Unique Minimal Normal Subgroup.
Theorem 1.1(Main Theorem)
If $G$ is a non cyclic finite group with a unique ...

0
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0
answers

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$G$ is a solvable group. Let $\pi(G)=\{p_{1}, p_{2}, p_{3}, p_{4}\}$
and $\{P_{1}, P_{2}, P_{3}, P_{4}\}$ be a Sylow basis of $G$. We have that $G=P_{1}P_{2}P_{3}P_{4}$. Set $H=P_{1}P_{2}P_{4}$ and $K=...

0
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0
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40
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The group U(2,2) has 16 real parameters. It contains the Lorentz group. What other spacetime groups does U(2,2) contain? How is it related to GL(4,R) which also has 16 real parameters?

1
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0
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119
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My question is the following: is there a (call it "canonical", "standard", or some other interesting and known) representation (probably reducible) of a finite group, which ...

8
votes

0
answers

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The infinite symmetric group $S_{\infty}$ of finitely supported permutations of $\mathbb{N}$ can be written as a colimit over the $S_n$'s with respect to the embedding $S_{n} \to S_{n+1}$ that maps $\...

27
votes

2
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529
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Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.

4
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0
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149
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a odd prime. We say that a pro-$p$ group has finite rank if it is isomorphic to a closed subgroup of $\GL_d(\mathbb{Z}_p)$ for some ...

1
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0
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144
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Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise:
(Exercise 5.9)
Let $G$ be a finite group and $N\unlhd G$, suppose ...

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0
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128
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Let $G$ be a finite group and $P \in \operatorname{Syl}_{p}(G)$. Let $z \in P \cap Z(N_{G}(P))$ be such that $tzt^{-1} \in P$, for some $t \in G$. Show that $tzt^{-1} = z$.
I have tried a lot solving ...

7
votes

2
answers

271
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Given a (compact) Lie group $G$, persumably disconnected, there exists a short exact sequence
$$1\rightarrow G_c\rightarrow G\rightarrow G/G_c\rightarrow 1$$
where $G_c$ is the normal subgroup which ...

3
votes

0
answers

210
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$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...

1
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0
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95
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$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime, $i$ a fixed positive integer and let $\Gamma_i$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^i\mathbb{Z}_p)$. ...

0
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0
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57
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Let $H$ be the discrete Heisenberg group. Is there an injective homomorphism $\varphi \colon H \to U(n)$ for some $n$? What about injective homomorphism $\varphi \colon H \to O(n)$ for some $n$?

1
vote

1
answer

80
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For a multidimensional subshift $X$ over $\mathbb Z^d$, the topological full group $[X]$ is the set of homeomorphisms $f$ of $X$ that can be written as $f : x \mapsto \sigma_{c(x)}(x)$ with $c : X \to ...

1
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0
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83
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Let $H$ be the discrete Heisenberg group, i.e., the set of matrices of the form
$\begin{bmatrix}
1 & x & z \\
0 & 1 & y \\
0 & 0 & 1
\end{bmatrix}$
where $x,y,z \in \mathbb{Z}$...

1
vote

1
answer

188
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Let $G =$ PGL$_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-...

9
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0
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85
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$\DeclareMathOperator\SL{SL}$Let $p$ be a prime, and let $\Gamma_r$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^r\mathbb{Z}_p)$.
Explicit formulas with formal ...

6
votes

0
answers

258
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Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...

1
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0
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30
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$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...

4
votes

0
answers

77
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In this question all groups are finite, and all spaces are nice (eg, simplicial sets).
Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of ...

1
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0
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207
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If we start with a perfect group $G$ of deficiency zero then there is a presentation $P$ of $G$ such that the number of relations and the number of generators for $P$ are the same. For such $P$, the ...

2
votes

1
answer

139
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Let $G$ be a compact (connected) semisimple Lie group. Let $G_\mathbb{C}$ be the complexification of $G$.
Is $G$ a maximal compact subgroup of $G_\mathbb{C}$?

1
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0
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61
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Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...

2
votes

0
answers

81
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Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...

11
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0
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158
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While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...

4
votes

1
answer

209
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Is there any perfect group which could be notated as $4^{11}\cdot M_{24}$ (a non-split extension of the largest Mathieu group by a homocyclic group of type $4^{11}$)?

1
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0
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99
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In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...

3
votes

1
answer

115
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For a generating set $S$ of a group $G$ denote by $\mathrm{Cay}(G,S)$ the corresponding Cayley graph.
For a finite graph $A$ denote by $\beta(A)$ its bandwidth.
Question: Has the "group bandwidth&...

2
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0
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108
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According to A complete classification of finite homogeneous groups, the finite simple groups $G$ which are homogeneous (meaning every isomorphism between subgroups extends to an automorphism of $G$) ...

7
votes

0
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Say that a group is rich if is contains isomorphic copies of all finite groups.
It is easy to produce rich groups, and also rich locally finite groups, for instance the restricted direct product $A=\...

0
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0
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128
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Is the matrix group generated by one copy each of the Coxeter reflection groups $B_3$ and $A_1I_2(8)$ intersecting in a copy of $A_1B_2$ (in the unique way this can be accomplished inside $GO(3)$, up ...

3
votes

1
answer

98
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(Followup to this question)
Consider a finite-dimensional Lie group $G$ and two conjugacy classes $H$ and $I$ of isomorphic subgroups of $G$.
Question. Is there some finite-dimensional Lie overgroup ...

2
votes

0
answers

47
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Let $A_\Gamma$ be an Artin group. The Salvetti complex $Sal(A_{\Gamma})$ can be briefly defined as the $2$-presentation complex associated to the usual presentation of the Artin group after attaching ...

2
votes

1
answer

165
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Consider a finite group $G$ and its complex group algebra $V_G$, on which $G$ acts. I would like to know: what are the polynomial $G$-invariants of $V_G$ i.e., the polynomial functions $p\in \mathbb{C}...

2
votes

1
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81
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Let $G$ be a topologically finitely generated pro-$p$-group. Assume that $G$ is torsion-free, it contains a dense free subgroup of infinite rank and the (topological) abelianization $G^{\text{ab}}$ of ...

0
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1
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109
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Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...

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0
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Let $w_{m}$ be the $m \times m$ matrix with ones on the non-principal diagonal and zeros elsewhere.
Let $V$ be the $2n+2$-dimensional quadratic space with the symmetric bilinear form $\left<,\right&...

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2
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129
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Is there a profinite group $G$ with a locally finite subgroup $H$ such that $\overline H$, the closure of $H$, is not torsion?

38
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2
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Does there exist a group $G$ such that
for any finite $K$ there is a monomorphism $K \to G$
for any $H$ with property 1 there is a monomorphism $G \to H$
If yes, is it the only one?

1
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0
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50
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The quasi-isolated elements of the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$ are classified in the paper "quasi-isolated elements in reductive groups" by C. Bonnafe.
Let's focus ...