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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6 votes
2 answers
259 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 ...
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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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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 ...
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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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4 votes
1 answer
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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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11 votes
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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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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 ...
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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 \...
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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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2 votes
1 answer
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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 $G$,...
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1 vote
1 answer
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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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7 votes
4 answers
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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}^...
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4 votes
0 answers
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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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4 votes
1 answer
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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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4 votes
2 answers
413 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 ...
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21 votes
2 answers
633 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 ...
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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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2 votes
1 answer
314 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 ...
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12 votes
2 answers
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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)=\...
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32 votes
3 answers
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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 ...
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8 votes
1 answer
249 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 $g_1^G\dotsb g_n^G$ ...
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1 vote
0 answers
152 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 ...
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20 votes
2 answers
893 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 \...
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  • 11k
2 votes
2 answers
584 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)}/{\...
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10 votes
2 answers
404 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 ...
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9 votes
2 answers
734 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 ...
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13 votes
2 answers
1k 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,\...
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  • 2,519
33 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 ...
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16 votes
4 answers
3k views

Is there an infinite group with exactly two conjugacy classes?

Is there an infinite group with exactly two conjugacy classes?
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7 votes
3 answers
791 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 ...
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7 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 ...
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12 votes
1 answer
555 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 ...
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12 votes
2 answers
1k views

A subgroup intersects every conjugacy class

For a subgroup $H$ of a given group $G$, I say $H$ is "big" if it has nonempty intersection with each conjugacy class of $G$. I have known that, trivially, $G$ itself is "big". And if $H$ is a normal ...
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