# Questions tagged [conjugacy-classes]

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### 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 ...
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### 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 $\mathrm{Cl}_g$ ...
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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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### A pair of non-conjugate subgroups: a simple proof

$\DeclareMathOperator\SO{SO}$Set \begin{aligned} \Gamma_1 &= \left\{ I_{6}, \; \gamma_1:= \left( \begin{smallmatrix} 0&1\\ 1&0 \\ &&0&1\\ &&1&0\\ &...
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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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### 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
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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. ...
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### $\mathrm{Sp}_n(q)$-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 ...
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### 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 $W$....
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### 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 ...
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### 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 ...
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1 vote
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### 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? ...
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### 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 ...
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