# Questions tagged [conjugacy-classes]

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26
questions

**2**

votes

**1**answer

123 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

61 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

112 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 ...

**2**

votes

**0**answers

55 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 ...

**0**

votes

**1**answer

202 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

399 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

81 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

181 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 $...

**3**

votes

**2**answers

289 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 ...

**20**

votes

**1**answer

500 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

53 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

253 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

385 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)=\...

**27**

votes

**3**answers

2k 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

236 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**

vote

**0**answers

144 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 ...

**20**

votes

**2**answers

765 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 \...

**1**

vote

**2**answers

492 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

358 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 ...

**8**

votes

**2**answers

613 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 ...

**13**

votes

**2**answers

859 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,\...

**32**

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 ...

**14**

votes

**4**answers

2k views

### Is there an infinite group with exactly two conjugacy classes?

Is there an infinite group with exactly two conjugacy classes?

**7**

votes

**3**answers

618 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 ...

**5**

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 ...

**11**

votes

**1**answer

504 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 ...