# Questions tagged [conjugacy-classes]

The conjugacy-classes tag has no usage guidance.

62
questions

15
votes

1
answer

835
views

### Algebraic structure on conjugacy classes

informally speaking, what algebraic structure does the set of conjugacy classes of a group carry?
Formally, I'm interested in natural operations on conjugacy classes. Let $\mathsf{Grp}$ be the ...

0
votes

1
answer

260
views

### Commutator group and conjugacy classes

Let $G$ be a finite solvable group which is not nilpotent, and let $H=[G,G]$ be the commutator subgroup of $G$. Does the following hold for $G$ and $H$?
"There exists $g \in G \setminus H$ and $h ...

0
votes

0
answers

68
views

### Extension of automorphism of shift of finite type

$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be two subshifts of finite type and $X\subset Y$, and $\phi:X\rightarrow X$ be a homeomorphism commuting with the shift map. Is there any homeomorphism $\...

4
votes

0
answers

110
views

### When is the intersection of cosets of a conjugacy class $0$-dimensional?

Let $G = \mathrm{SL}_n$ (say); let $K$ be a field. Let $g$ be a regular semisimple element of $G(K)$, and $\mathrm{Cl}_g$ its conjugacy class, considered as an algebraic variety. Then $\mathrm{Cl}_g$ ...

7
votes

1
answer

349
views

### 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 ...

4
votes

1
answer

286
views

### 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\\
&...

0
votes

1
answer

197
views

### 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 ...

3
votes

1
answer

101
views

### Are isomorphic maximal tori stably conjugate?

Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...

1
vote

0
answers

182
views

### 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. ...

4
votes

1
answer

365
views

### 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{...

2
votes

2
answers

282
views

### Number of conjugacy classes of a semi-direct product of two finite groups

Let $G$ and $H$ be two finite groups. Let $r(G)$ be the order of the set of conjugacy classes of $G$. We know $$r(G\times H)=r(G)\times r(H).$$ My problem is: if there is a semi-direct product $G\...

10
votes

2
answers

700
views

### Conjugacy classes in towers of groups

Let $\Gamma$ be a group and $\Gamma_1\supset\Gamma_2\supset\dots$ subgroups of finite index, such that $\bigcap_{j=1}^\infty \Gamma_j=\{1\}$.
Let $1\ne\gamma\in\Gamma$ and let $[\gamma]=[\gamma]_\...

3
votes

1
answer

196
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 ...

1
vote

0
answers

49
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 ...

2
votes

0
answers

114
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
votes

1
answer

165
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 $...

5
votes

0
answers

205
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 ...

1
vote

0
answers

60
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{...

12
votes

2
answers

340
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. (...

2
votes

0
answers

131
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 ...

1
vote

1
answer

240
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 $2$-...

3
votes

1
answer

109
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 ...

7
votes

1
answer

435
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 ...

2
votes

1
answer

120
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 ...

7
votes

2
answers

433
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

157
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 ...

2
votes

1
answer

95
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 ...

4
votes

2
answers

664
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

192
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 ...

11
votes

1
answer

590
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 ...

7
votes

2
answers

530
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 \...

1
vote

0
answers

143
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 ...

1
vote

0
answers

115
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 \...

1
vote

2
answers

404
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)...

3
votes

1
answer

535
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 $G$,...

1
vote

1
answer

77
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 $E_7(q)$?

4
votes

0
answers

154
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 ...

4
votes

0
answers

176
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 ...

1
vote

1
answer

247
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 $p$-...

7
votes

4
answers

1k
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}^...

4
votes

0
answers

98
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 ...

4
votes

1
answer

309
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 $W$....

4
votes

2
answers

531
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 ...

21
votes

2
answers

675
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 ...

1
vote

0
answers

66
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
votes

1
answer

343
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 ...

12
votes

2
answers

482
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)=\...

12
votes

1
answer

492
views

### 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 ...

34
votes

3
answers

5k
views

### 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

256
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 $g_1^G\dotsb g_n^G$ ...