Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,942
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Word maps from $\mathrm{Cl}^{n+1}$ to $G^n$: quasi-injectivity?
Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same ...
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0
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36
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Separability in Coxeter groups
I am looking for a reference for the following statement:
Theorem. Let $\Gamma$ be a finite labelled graph and $C(\Gamma)$ the corresponding Coxeter group. For every $\Xi \subset V(\Gamma)$, the ...
- 7,511
8
votes
3
answers
678
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Finding a nilpotent, infinite, f.g., virtually abelian, irreducible integer matrix group
I was wondering if anyone here knows of an example of a group $M \leq \mathrm{GL}_n(\mathbb{Z})$ which is
nilpotent,
infinite,
finitely generated,
virtually abelian,
irreducible (over $\mathbb{Z}$ or ...
- 1,113
4
votes
2
answers
442
views
Is every bounded representation of Z unitarisable when all sets are measurable?
For the purpose of this question, a group is amenable iff there exists a Følner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded ...
- 60.4k
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0
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59
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Connection between certain finite groups and Frobenius algebras
This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition.
Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...
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1
vote
0
answers
51
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Equivalent definition of Spin group in terms of automorphisms
Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...
- 1,871
16
votes
3
answers
904
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Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 2,052
1
vote
1
answer
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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 ...
CommunityBot
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votes
0
answers
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What can we say about the average order of group members of alternating group?
During an ongoing research I dealt with the concept of orders of group members. The following question remained a gap in my analysis. Any insight is appreciated.
Let $ \bar{o}(G) $ be the average ...
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2
votes
1
answer
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Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
CommunityBot
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2
answers
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When are (Abelian) Cayley graphs also expanders?
I want to ask the question in two parts,
(1)
Is there some fundamental distinguishing property between Abelian and non-Abelian Cayley graphs? (say some specific proof technique which distinguishes ...
- 175
4
votes
1
answer
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The number of irreducible characters of simple groups of Lie type
Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$.
Set $\mathrm{C}_{S}(\sigma)$ the ...
- 42.8k
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1
answer
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Commutator subgroup of $\mathrm{GL}_n(R)$ when $R$ is a PID with unity
Hua and Reiner in their paper titled "Automorphisms of the unimodular group" have established what will be the commutator subgroup of $\mathrm{GL}_n(\mathbb{Z})$, $\forall n\geq 0$. I have ...
- 7,463
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4
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Prove these are not surface groups
For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation:
$$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$
For $n = 1$, ...
- 7,463
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1
answer
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Normal closure of $e_{12}$ in the congruence subgroup $\Gamma_1(p)\subset \mathrm{SL}_2(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$For an odd prime $p$, let
$$\Gamma_1(p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in \SL_2(\mathbb{Z}):\begin{pmatrix}a & b \\ c & d\end{pmatrix}\...
- 43.6k
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0
answers
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Equation in a nilpotent group
Let $G$ be a nilpotent group of class at most $r$
(that is, $\gamma^{r+1}G=1$).
Let elements $g_1,\dotsc,g_n\in G$ be fixed.
We are interested in the set $V\subseteq\mathbb Z^n$ of solutions $x=(x_1,\...
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2
answers
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Computing the Abelian invariants of a subgroup of a f.g. Abelian group
We have a f.g. Abelian group $A$ given as a direct sum of $N$ cyclic subgroups $C_{k_j}=\langle x_j\rangle$, with $k_j\in \{2,\dots,\infty\}$, $1\leq j \leq N$, and the associated homomorphism $\phi:\...
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Friedrich Schur on the BCHD theorem (notes in English)
According to Sternberg in his book Lie Algebras,
The formula [the BCHD formula] is named after three mathematicians, Campbell, Baker, and
Hausdorff. But this is a misnomer. Substantially earlier than ...
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answer
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Number of conjugacy classes of pairs of commuting elements II
This post follows up on a discussion initiated in Number of conjugacy classes of pairs of commuting elements.
Consider a finite group $G$ and let $r_G$ represent the number of conjugacy classes of ...
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How can I get my hands on McKay's "Finite p-groups" lecture notes?
How can we find Susan McKay's "Finite $p$-groups" lecture notes?
The notes I'm talking about are these.
I emailed Peter Cameron, but he has since moved to a different university, and has no ...
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I am looking for "Finite $p$-group" notes of Susan McKay, where can I find them? [closed]
A teacher of mine suggested me to read finite $p$-group lecture notes of Susan McKay, but I can't find them anywhere. Can somebody help me out?
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How to think about the simple reflection $s_0$ in the affine Weyl group?
Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
- 2,758
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votes
2
answers
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Simple connectedness of Levi subgroup
Let us consider a connected and simply connected semisimple algebraic group $G$ over $\mathbb{C}$, $B$ a Borel subgroup and $T$ a maximal torus contained in $B$.
Let $P_1$, $P_2$ be two standard ...
- 11.4k
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votes
1
answer
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Holt's Theorem on doubly transitive groups with $2$-central involutions fixing only one letter
In D. F. Holt, Transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes a unique point, Proc. London Math. Soc. 37 (1978), 165–192, Derek Holt classifies finite doubly ...
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A malnormal embedding theorem?
Let $Q$ be a recursively presented group. Is it possible to embed $Q$ into a finitely presented group $G$ such that the image of $Q$ is malnormal in $G$?
Note that a subgroup $H$ of $G$ is malnormal ...
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2
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What is a cogroup and what are coactions?
What is a cogroup and what are coactions?
A very nice way to think about a group action on an object $X$ is as a group homomorphism from $G$ to $\operatorname{Aut}(X)$. Is there something similar for ...
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1
answer
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Loop manipulation subgroup of the braid group
Recently, I came across a subgroup of the braid group $B_{2n}$ that I'm calling the "loop manipulation" group $H_n$.
The idea is that we treat pairs of adjacent strands in the braid group as ...
- 26.4k
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0
answers
117
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What is this quotient of the free product?
Previously asked at MSE. The construction here can generalize to arbitrary algebras (in the sense of universal algebra) in the same signature with the only needed tweak being the replacement of "...
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Special groups, special resolutions and group cohomology
$\newcommand{\Z}{\mathbf{Z}}$ Let $G$ be a non-abelian group. And let $\Z$ be the ring of integers.
Under which condition on the group $G$ can we find a free resolution $F_{\bullet}\rightarrow \Z$
of $...
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1
answer
328
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How bad can the recursive properties of finitely presented groups be?
Any finitely presented group naturally gives rise to an edge-labeled graph (the Cayley graph) and I am considering paths through this graph. Paths correspond to infinite sequences of generators, so ...
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Failure of Tomiyama's property ($F$) for reduced group $C^*$-algebras
Are there known examples of discrete groups such that the minimal tensor product of their reduced group $C^\ast$-algebras does not have Tomiyama's property ($F$)?
Such groups must necessarily be non-...
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1
answer
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When the fundamental group of subgraph of groups embeds?
Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
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1
answer
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Automorphism group of tensor product of two graphs
Is there any relation between the automorphism group of the tensor product of two graphs $G = G_1 \times G_2$ and the automorphism groups of $G_1$ and $G_2$?
I am aware of the nice results for the ...
CommunityBot
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1
answer
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Examples of Noetherian integral group ring
I need to study the integral group ring of the fundamental group of a manifold. My knowledge of group and ring theory is very limited. I am looking for some examples of groups $G$ for which $\Bbb ZG$ ...
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In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?
Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
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Groups $P$ of order $p^5$ with $\Omega_1(P)=P$
I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...
- 60.4k
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A pair of non-conjugate subgroups: a simple proof
$\DeclareMathOperator\SO{SO}$Set
\begin{equation}
\begin{aligned}
\Gamma_1 &=
\left\{
I_{6},
\;
\gamma_1:=
\left(
\begin{smallmatrix}
0&1\\
1&0 \\
&&0&1\\
&&1&0\\
&...
- 11.4k
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1
answer
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Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
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votes
1
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Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? [closed]
I asked this question on MSE here
This question was inspired by: The influence of conjugacy class sizes on the
structure of finite groups.
My question is as follows: Is there a way to study the ...
- 29.8k
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votes
0
answers
56
views
Finite $p$-groups of maximal class whose generators have order $p$
Let $G$ be a finite $p$-group of maximal nilpotency class that is not cyclic of order $p^2$. Then $G$ is $2$-generated, say $G=\langle a,b\rangle$. Is there a classification in the case when $a^p=b^p=...
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votes
1
answer
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n-ary (polyadic) group "defined for tuples"
Consider an n-ary (polyadic) group, which is a generalization of the "usual" ("binary") groups to n-tuples. The definition requires appropriate standard generalizations of ...
CommunityBot
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Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
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Subgroup of p-adic units
Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
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answers
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On the conjugation action on the first congruence subgroup of special linear group over $p$-adic fields
Let $\mathcal{O}_{L}$ be the ring of integers of a finite extension $L$ of $p$-adic number fields $\mathbb{Q}_{p}$ where $p$ is an odd prime. Let $\mathfrak{m}_{L}$ be the maximal ideal of $\mathcal{O}...
- 491
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votes
1
answer
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Finite-maximal subgroups of orthogonal groups
I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite.
My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
- 42.8k
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votes
0
answers
90
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Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
- 13.6k
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votes
1
answer
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Is the group of translations of an affine plane always commutative?
$\DeclareMathOperator\Dil{Dil}\DeclareMathOperator\Trans{Trans}\DeclareMathOperator\Col{Col}$An affine plane is a set of points $X$ endowed with a family $\mathcal L$ of subsets of $X$, called lines, ...
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0
answers
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Are all these groups CAT(0) groups?
Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive curvature ...
- 1,446
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votes
2
answers
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On the cohomology ring of the Grassmannian
The basis of Schubert classes for the cohomology ring $H^*(\text{Gr}(m,N))$ of the Grassmannian of $m$-dimensional subspaces of $\mathbb{C}^N$ is indexed by $L(m,N-m)$, the poset of all partitions ...
- 1,446
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vote
0
answers
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Finitely presentable groups are residually finite if and only if they are universally pseudofinite
Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
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