Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,942
questions
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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
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votes
0
answers
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 ...
- 26.1k
8
votes
3
answers
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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 ...
- 5,194
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0
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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 ...
- 19.4k
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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
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0
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76
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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 ...
- 591
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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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votes
1
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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}\...
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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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4
votes
1
answer
220
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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 ...
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views
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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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 ...
- 26.1k
5
votes
2
answers
272
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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:\...
- 13.7k
3
votes
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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 ...
- 4,353
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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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5
votes
2
answers
282
views
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 ...
- 541
1
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2
answers
433
views
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 ...
user525442
5
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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 ...
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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 ...
- 647
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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 "...
- 19.2k
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views
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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answer
140
views
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 ...
- 1,216
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votes
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$ ...
- 1,087
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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 ...
- 143
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answer
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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\\
&...
- 2,321
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votes
1
answer
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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 ...
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answer
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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$. ...
- 573
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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=...
2
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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}...
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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 ...
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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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0
answers
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views
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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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$ ...
- 1,216
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votes
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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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Which groups can be generated by a single conjugacy class?
How can we characterize the finite groups generated by a subset of a single conjugacy class?
This post asks for well-known families of finitely generated groups generated by a single conjugacy class. ...
- 43
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vote
0
answers
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Bounds for the orders of second largest subgroups of $\mathrm{SL}_n(\mathbb F_q)$
$\DeclareMathOperator\SL{SL}$By Patton's thesis, except for a finite number of possibilities, the $(n-1, 1)$ parabolic subgroup, $P$ say, has the largest number of elements among all non-trivial ...
- 1,362
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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}...
- 143
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votes
2
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Order of abelian subgroup of the automorphism group of an abelian group
Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
- 1,216
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votes
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answers
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Implementation of the nerve of a category in GAP
I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...
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Interpretation of Kazhdan T property cohomologically
$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology.
In general, we heuristically have $H^1(G,Ad(V))$ (...
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votes
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answer
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Number of conjugacy classes of pairs of commuting elements
Let $G$ be a finite group and denote by $r_G$ the number of conjugacy classes of pairs of commuting elements, i.e. the cardinality of the following set $$ A_G = \{ c(a_1,a_2) \ | \ a_1,a_2 \in G \text{...
- 25.9k
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views
How small can maximal subgroups be?
Given a finite group $G$, let $p(G)$ denote the number of prime factors
of the order of $G$ (counting multiplicities).
Does there exist a function $f: \mathbb{N} \rightarrow \mathbb{N}$
which grows ...
- 19.5k
3
votes
0
answers
104
views
Finite approximability of graphs with finitely many automorphisms
In this question, all graphs are understood to be simple and undirected, and have countably many vertices and edges, but not necessarily finite.
Let $G = (V, E)$ be a graph. It is clear that any ...
- 1,262
2
votes
1
answer
160
views
Order of a loop around a cone point
Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
- 533
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votes
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answers
94
views
On the irreducible submodules of adjoint representations $\text{ad}^{0}$
Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is ...
- 491
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vote
0
answers
127
views
Isomorphic quotients of a countably infinitely-generated free abelian group
Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
- 11
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votes
0
answers
49
views
Approximating open subset of profinite group by union of cosets of ideal
I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
- 1
4
votes
1
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Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
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votes
1
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Extending primitive systems in free groups
It is a version of a question I asked on Math.StackExchange (no answers yet). Suppose $a_1,a_2,\ldots,a_n,b$ is a primitive system in a non-abelian free group $F$ (a part of a basis of $F$). Let $c \...
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Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
- 23.9k