Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

Filter by
Sorted by
Tagged with
2 votes
1 answer
67 views

Schur multiplier of $\mathrm{SL}(2,\mathbb{Q})$

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 views

Direct Product of Finite Groups [closed]

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 views

A group-theoretic lemma in a paper by Ershov and He

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

Finite dimensional unitary representations of the discrete Heisenberg group

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 votes
1 answer
312 views

Why is $\mathrm{SO}(4)$ not a simple Lie group?

$\DeclareMathOperator\SO{SO}$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 $A_4$ ...
  • 109
3 votes
1 answer
108 views

Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$

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 answers
93 views

The property of self-normalizing subgroup

$G$ 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 views

Suzuki-Ree Lie algebras

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 answers
148 views

Commutation classes of reduced decompositions of the longest element of the Weyl group with one element

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 views

Centre of orbifold fundamental group of torus (Klein bottle) with one cone point

$\newcommand{\orb}{\mathrm{orb}}$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, $$\...
  • 365
1 vote
0 answers
65 views

Generating set of a group with a unique minimal normal subgroup

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 ...
  • 11
0 votes
0 answers
89 views

$G=HK$, $N_H(P_4)=P_4$ and $N_K(P_4)=P_4$

$G$ 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 votes
0 answers
40 views

What spacetime symmetry groups does U(2,2) contain?

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 vote
0 answers
119 views

Representation of a group with dimension equal to a number of conjugacy classes

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
210 views

Colimits of symmetric groups

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 answers
529 views

Is there a flat manifold with trivial first homology?

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 votes
0 answers
149 views

When a pro-$p$ group of finite rank can be embedded into the first congruence subgroup of ${\rm GL}_{N}(\mathbb{Z}_{p})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$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 vote
0 answers
144 views

Character extension about $Q_8$

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 ...
-2 votes
0 answers
128 views

Intersection of a Sylow subgroup with the center of its normaliser [closed]

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 views

Injectivity of the cohomology map induced by some projection map

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 views

Representations of triangle groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$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,...
  • 119
1 vote
0 answers
95 views

Generators of of $p$-adic congruence subgroups of $\operatorname{SL}_2$

$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime, $i$ a fixed positive integer and let $\Gamma_i$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^i\mathbb{Z}_p)$. ...
  • 451
0 votes
0 answers
57 views

Faithfull compact representation of the discrete Heisenberg group

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 views

Topological full groups of subshifts: differences between one-dimensional and multi-dimensional subshifts

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 ...
  • 113
1 vote
0 answers
83 views

Classification of the normal subgroups of the discrete Heisenberg group

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 views

Kronecker product preserves the conjugacy relation?

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 votes
0 answers
85 views

Derived subgroups of 2-adic congruence subgroups of $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}$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 views

(CFSG-free) Finite simple groups whose character degrees square divide its order

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 vote
0 answers
30 views

What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?

$\newcommand{\Lie}{\operatorname{Lie}}$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 ...
  • 161
4 votes
0 answers
77 views

When can the trace on cohomology be computed as the Euler characteristic of fixed points?

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 ...
  • 1,597
1 vote
0 answers
207 views

Presentation complexes with same homology and different fundamental groups

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 ...
  • 159
2 votes
1 answer
139 views

Question about maximal compact subgroups of Lie groups

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 $G_\mathbb{C}$?
  • 99
1 vote
0 answers
61 views

Another matrices for a semigroup with intermediate growth

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 views

Question concerning relationships between different $p$-modular systems and Brauer character values

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 votes
0 answers
158 views

On an Artin (?) subgroup of braid groups

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 views

Does a perfect $4^{11}\cdot M_{24}$ exist?

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 vote
0 answers
99 views

Embedding (Kronecker product) preserves the structure?

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 views

Bandwidth of finite groups

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&...
  • 31
2 votes
0 answers
108 views

Homogeneity of finite simple groups

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 answers
191 views

Locally finite groups containing all finite groups

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=\...
  • 54k
0 votes
0 answers
128 views

Is this the free product with amalgamation?

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 views

Fusing conjugacy classes II

(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 views

Chain complex of the Salvetti complex of an Artin group

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 ...
  • 131
2 votes
1 answer
165 views

Invariants of the group algebra of a finite group

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 answer
81 views

Nontrivial abelianization of torsion-free pro-$p$-group which contains a dense free subgroup is infinite?

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 votes
1 answer
109 views

Intersection of identity components

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})^{\...
1 vote
0 answers
60 views

Question on the representative of the longest Weyl element of $\mathrm{SO}(2n+1)$

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&...
  • 1,535
1 vote
2 answers
129 views

Looking for an example of profinite groups

Is there a profinite group $G$ with a locally finite subgroup $H$ such that $\overline H$, the closure of $H$, is not torsion?
38 votes
2 answers
3k views

Is there a smallest group containing all finite groups?

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 vote
0 answers
50 views

Centralisers of involutions not quasi-isolated

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 ...

1
2 3 4 5
147