# Questions tagged [conjugacy-classes]

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### Conjugacy classes of $\mathrm{SL}_n(\mathbb{Z})$ with $n > 2$

For $\mathrm{SL}_2(\mathbb{Z})$, there is a concise description of all conjugacy classes via several families of words of a certain form, as e.g. described in an answer to this post. In particular, in ...
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### 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 ...
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### 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 ...
1 vote
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### 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 ...
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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 ...
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### 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 ...
159 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 ...
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### 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 ...
1 vote
### 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 ...