# Questions tagged [conjugacy-classes]

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### 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 ...
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### 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. (...
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### 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
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### 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 $2$-...
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### 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 ...
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### 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 ... 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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)...
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### 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
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### 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 $E_7(q)$?
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### 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 ...
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### 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 ...
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### 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 $p$-...
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### 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 ...
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### Is there an infinite group with exactly two conjugacy classes?

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