Questions tagged [conjugacy-classes]
The conjugacy-classes tag has no usage guidance.
51
questions
3
votes
1
answer
169
views
Finite subgroup of $\operatorname{Sp}(2n,K)$
Let $G$ be the algebraic group $\operatorname{Sp}(2n, K)$ where $K$ is an algebraically closed field of characteristic not $2$. There is a quaternion subgroup $Q$ such that $Q/Z(G)$ is elementary ...
- 449
0
votes
0
answers
42
views
A question on width vs covering of the subgroup generated by a conjugacy class in a finite group
Let $G$ be a finite group and $C$ be a conjugacy class of $G$. It is clear that there exists $k\in \mathbb{N}$, such that $1\cup C\cup C^2 \cup \cdots \cup C^k=\langle C \rangle$. Note that $\langle C ...
- 420
2
votes
0
answers
105
views
Elementary abelian 2-subgroups of $\mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ (with and without choice)
Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$.
As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are ...
- 4,003
4
votes
1
answer
154
views
A probability problem in the conjugacy classes of symmetric group
Assume that $\sigma\in S_n$ has the cycle type $(p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $...
- 103
0
votes
0
answers
82
views
Is there a formula for $fgf^{-1}$ in $G_2$?
Let $G_2$ be the exceptional Lie group of smallest dimension, represented as the exponentials of the antisymmetric real $7×7$ matrices $A$ with $$\sum_{i=1}^{3}{A(x+2^i,x-2^{i-1})} = 0$$ for integer $...
- 321
3
votes
0
answers
150
views
Almost conjugate subgroups of compact simple Lie groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
- 2,269
1
vote
0
answers
59
views
Choice of generators to make the centralisers connected
In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...
- 449
12
votes
2
answers
321
views
Conjugacy classes as left Kan extension of forgetful functor
Let $\mathbf{Set}$, $\mathbf{Grp}$, and $\mathbf{Grp}^{\rm conj}$ denote the categories of sets and functions, groups and homomorphisms, and groups and homomorphisms up to conjugation, respectively. (...
- 14.7k
2
votes
0
answers
120
views
Need for "minimal representation" of a symmetric group
I need to construct a representation of a symmetric group $S_n$, in which a character of the conjugacy class $(n)$ (a class of permutations, which are cycles of a maximal possible length $n$) would be ...
- 31
1
vote
1
answer
212
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 $2$-...
- 449
3
votes
1
answer
105
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 ...
- 2,593
6
votes
1
answer
416
views
Center of a monoid ring
According to the Wikipedia page the center of a group ring $R[G]$ is the set:
$$
\{ p | \forall g,\, h \in G.\, p(g) = p(hgh^{-1}) \}
$$
i.e. class functions which do not distinguish elements of the ...
user491484
2
votes
1
answer
112
views
Zeroes of characters of general linear group induced from certain characters of parabolic subgroups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which ...
- 21
7
votes
2
answers
342
views
Size of conjugacy classes in infinite groups
Let G be an infinite group wich is finitely generated.
Is that true that the size of all finite conjugacy classes is bounded?
What I know. If G is a finitely generated FC-group then it's true (follows ...
2
votes
0
answers
115
views
Conjugacy class of upper triangular matrices over algebraically closed field: Reference request
We know that the conjugacy classes of $A\in M_n(\mathbb{C})$ are determined by the characteristic polynomial of $A$ and a partition of $n$. Is there an analogous statement for upper triangular ...
- 215
2
votes
1
answer
77
views
conjugacy in adjoint representation
Let $G$ be an adjoint algebraic group over $\mathbb{C}$, $\mathfrak{g}$ its Lie algebra.
Let $\rho:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let $g,g'\in G$ be two semisimple ...
- 3,314
4
votes
2
answers
640
views
Conjugacy classes in the automorphism group of a simple Lie algebra
A lower bound of the number of conjugacy classes in the automorphism group of a simple Lie algebra $\mathfrak{s}$, of finite dimension over an arbitrary field $\mathbb{F}$, can be the size of the ...
4
votes
1
answer
181
views
Lower bound on size of largest conjugacy class of centreless perfect group
Problem 20.30 in the Kourovka Notebook asks whether the maximum size
of a conjugacy class of a perfect and centreless finite group $G$ is bounded below
by $|G|^{\frac{1}{2}}$. Clearly, there cannot be ...
- 18.9k
11
votes
1
answer
562
views
What are the conjugacy classes of the category of ($\kappa$-small) sets?
$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such ...
- 9,565
7
votes
2
answers
457
views
Groups with three conjugacy classes that define an ordering
Consider the following property for a group $(\mathcal{G},\cdot,1)$:
There are exactly three conjugacy classes $\{1\}$, $\mathcal{C}_1$, $\mathcal{C}_2$ in $\mathcal{G}$, and we have $\mathcal{C}_1 \...
- 2,307
1
vote
0
answers
126
views
How many conjugacy classes of cyclic subgroups of order $p^2$ does $\operatorname{GL}_{n}(\Bbb Z / p\Bbb Z)$ have?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Let $f\in \Hom((\Bbb Z/p^2\Bbb Z),\GL_{n}(\Bbb Z / p\Bbb Z))$ be an injective homomorphism. What is the number ...
- 461
1
vote
0
answers
102
views
Haar measure decomposition using orbital integrals
Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \...
- 680
1
vote
2
answers
364
views
Are the character degrees determined by the conjugacy class sizes?
The computation below (part 1) shows that if two finite groups of order at most $100$ have the same (ordered) list of conjugacy class sizes, then they also have the same (ordered) list of (irreducible)...
- 25.1k
2
votes
1
answer
453
views
Conjugacy of elements in a parabolic subgroup
Let $G$ be a complex connected reductive group, and let $P \subseteq G$ be a parabolic subgroup. My question is the following: if $g$ and $h$ are elements of $P$ which are conjugate as elements of $G$,...
- 481
1
vote
1
answer
76
views
What do conjugacy classes of involutions like in finite simple group $E_7(q)$?
Are there any refences for conjugacy classes of involutions in finite simple group $E_7(q)$?
- 261
4
votes
0
answers
147
views
Reference request - conjugacy classes over local fields
Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in ...
- 5,412
4
votes
0
answers
167
views
rational representants of sigma-conjugacy classes
Let $G$ be a connected reductive group over a local non-archimedean field $K$. Let $\widehat{K}^{nr}$ be the completion of the maximal unramified extension of $K$ and let $\sigma$ denote the Frobenius ...
- 735
0
votes
1
answer
236
views
Number of commuting pairs in p-group
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$ having
entries of $1$ on the diagonal. The set $U$ is a Sylow $p$-...
- 171
7
votes
4
answers
782
views
Maximum conjugacy class size in $S_n$ with fixed number of cycles
Context: It is well known that given a permutation in $S_n$ with $a_i$ $i$-cycles (when written as a product of disjoint cycles), the size of the conjugacy class is given by
$$ \frac{n!}{\prod_{j=1}^...
- 211
4
votes
0
answers
94
views
$\mathrm{Sp}_n(q)$-conjugacy classes in $\mathrm{GL}_{2n}(q)$
The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in ...
- 287
4
votes
1
answer
286
views
Coxeter groups generated by one finite conjugacy class
Let $(W,S)$ be an arbitrary Coxeter system. We consider the following scenario:
Let $\mathcal{O}$ be a conjugacy class of an element $w$ in $W$ which is finite and which generates the whole group $W$....
- 974
4
votes
2
answers
467
views
Variety of conjugacy classes
Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
- 1,190
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votes
2
answers
661
views
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?
Let $p$ be a prime; $\mathbb{F}_{p}$ is the field with $p$ elements
and $\mathbb{F}_{p}[t]$ the ring of polynomials in $t$ over $\mathbb{F}_{p}$.
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the ...
- 3,713
1
vote
0
answers
63
views
Relation Among Conjugacy Classes
This is more a request to find out if there is any work in the literature
discussing certain things.
Is there a naturally defined partial ordering on the set of conjugacy classes of a finite group G? ...
- 2,483
2
votes
1
answer
333
views
Torsion-free groups with finite conjugacy classes
Does there exist a finitely presented, torsion-free group $G$ which has conjugacy classes of finite size greater than one?
This condition came up in a research project, and we would like to rule out ...
- 34.6k
12
votes
2
answers
439
views
Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?
Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\...
- 492
12
votes
1
answer
452
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Constructing the largest finite group with a fixed number of conjugacy classes
It is known that there are finitely many finite groups with a given number of conjugacy classes. How can one construct (or get a character table for) the groups $G$ that realize the maximum possible ...
- 2,673
34
votes
3
answers
4k
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Conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$
I was wondering if there is some description known for the conjugacy classes of $$\mathrm{SL}_2(\mathbb{Z})=\{A\in \mathrm{GL}_2(\mathbb{Z})|\;\;|\det(A)|=1\}.$$ I was not able to find anything about ...
8
votes
1
answer
254
views
Need a good name for an algorithmic problem in groups that generalizes the conjugacy problem
I am looking for a good name for the following problem:
Given elements $g_1,\dotsc,g_n$ in a (finitely generated) group $G$, determine if the product of their conjugacy classes $g_1^G\dotsb g_n^G$ ...
- 1,400
1
vote
0
answers
160
views
Conjugacy scheme, fppf versus GIT
I would be glad to have some guidance in the following.
Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
- 5,412
21
votes
2
answers
968
views
Is there a big solvable subgroup in every finite group?
Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
- 11.2k
2
votes
2
answers
633
views
Rational Conjugacy Classes of Finite Groups
Suppose $G$ is a finite group and $A$ is the set of all character values of $G$. By character values, I mean entries of the character table of $G$. Let $\Gamma = \operatorname{Gal}({\mathbb{Q}(A)}/{\...
10
votes
2
answers
432
views
existence of a finite group which is the union of self normalizing subgroups
Can a finite group G be the union of self normalizing subgroups such that the intersection between any two of these subgroups is equal to the unit of the group G? I don't think so but I can't prove ...
- 101
9
votes
2
answers
812
views
What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?
Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams.
Question: To what extent is this ...
- 22.7k
13
votes
2
answers
1k
views
The Simultaneous Conjugacy Problem in the symmetric group $S_N$
We are interested in the following notions in the case $G=S_N$, the symmetric group on
$\{1,\dots,N\}$.
Fix a group $G$ and a number $d$. For $(g_1,\dots,g_d)\in G^d$ and $x\in G$, define
$$(g_1,\...
- 2,968
33
votes
2
answers
2k
views
Examples of finite groups with "good" bijection(s) between conjugacy classes and irreducible representations?
For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ...
- 22.7k
16
votes
4
answers
3k
views
Is there an infinite group with exactly two conjugacy classes?
Is there an infinite group with exactly two conjugacy classes?
- 343
7
votes
3
answers
830
views
Characters of p-groups
Berkovich mentioned the following result of Mann in his book on p-groups:
The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1.
Do you know any reference for ...
- 307
8
votes
4
answers
2k
views
Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?
Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with ...
- 22.7k
12
votes
1
answer
580
views
Why would dim primitive irrep divide size of some conjugacy class ?
From Isaacs et.al. 2005
Conjecture C. Let χ be a primitive
irreducible character of an arbitrary
finite group G. Then χ(1) divides |
clG(g)| for some element g ∈ G.
Here, of course, we ...
- 22.7k