Questions tagged [conjugacy-classes]
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54
questions
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131
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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. ...
3
votes
1
answer
270
views
Number of conjugacy classes of pairs of commuting elements
Let $G$ be a finite group and denote by $r_G$ the number of conjugacy classes of pairs of commuting elements, i.e. the cardinality of the following set $$ A_G = \{ c(a_1,a_2) \ | \ a_1,a_2 \in G \text{...
2
votes
2
answers
217
views
Number of conjugacy classes of a semi-direct product of two finite groups
Let $G$ and $H$ be two finite groups. Let $r(G)$ be the order of the set of conjugacy classes of $G$. We know $$r(G\times H)=r(G)\times r(H).$$ My problem is: if there is a semi-direct product $G\...
10
votes
2
answers
691
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Conjugacy classes in towers of groups
Let $\Gamma$ be a group and $\Gamma_1\supset\Gamma_2\supset\dots$ subgroups of finite index, such that $\bigcap_{j=1}^\infty \Gamma_j=\{1\}$.
Let $1\ne\gamma\in\Gamma$ and let $[\gamma]=[\gamma]_\...
3
votes
1
answer
190
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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 ...
1
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0
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48
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A question on width vs covering of the subgroup generated by a conjugacy class in a finite group
Let $G$ be a finite group and $C$ be a conjugacy class of $G$. It is clear that there exists $k\in \mathbb{N}$, such that $1\cup C\cup C^2 \cup \cdots \cup C^k=\langle C \rangle$. Note that $\langle C ...
2
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0
answers
112
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Elementary abelian 2-subgroups of $\mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ (with and without choice)
Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$.
As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are ...
4
votes
1
answer
159
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A probability problem in the conjugacy classes of symmetric group
Assume that $\sigma\in S_n$ has the cycle type $(p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma_1$ and $\sigma_2$ are chosen uniformly in the conjugacy class of $...
4
votes
0
answers
186
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Almost conjugate subgroups of compact simple Lie groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SO{SO}$Let $G$ be a compact connected Lie group.
Definition:
Two finite subgroups $H_1,H_2$ of $G$ are said to be almost ...
1
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0
answers
60
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Choice of generators to make the centralisers connected
In $G=\operatorname{PGL}_{2n}(\textbf{C})$, WLG, we assume all the toral elementary abelian 2-subgroups in discussion are in $T$, the image in $G$ of the group of diagonal matrices in $\operatorname{...
12
votes
2
answers
332
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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. (...
2
votes
0
answers
130
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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
1
answer
232
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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$-...
3
votes
1
answer
108
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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 ...
7
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1
answer
429
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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 ...
2
votes
1
answer
114
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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 ...
7
votes
2
answers
387
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 ...
2
votes
0
answers
142
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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 ...
2
votes
1
answer
84
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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 ...
4
votes
2
answers
657
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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 ...
4
votes
1
answer
188
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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 ...
11
votes
1
answer
570
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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 ...
7
votes
2
answers
518
views
Groups with three conjugacy classes that define an ordering
Consider the following property for a group $(\mathcal{G},\cdot,1)$:
There are exactly three conjugacy classes $\{1\}$, $\mathcal{C}_1$, $\mathcal{C}_2$ in $\mathcal{G}$, and we have $\mathcal{C}_1 \...
1
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0
answers
139
views
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 ...
1
vote
0
answers
110
views
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 \...
1
vote
2
answers
383
views
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)...
2
votes
1
answer
496
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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
1
answer
77
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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)$?
4
votes
0
answers
152
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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 ...
4
votes
0
answers
171
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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 ...
0
votes
1
answer
240
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
1k
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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}^...
4
votes
0
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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 ...
4
votes
1
answer
296
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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 $W$....
4
votes
2
answers
507
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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 ...
21
votes
2
answers
667
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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 ...
1
vote
0
answers
66
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
341
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
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2
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475
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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)=\...
12
votes
1
answer
484
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Constructing the largest finite group with a fixed number of conjugacy classes
It is known that there are finitely many finite groups with a given number of conjugacy classes. How can one construct (or get a character table for) the groups $G$ that realize the maximum possible ...
34
votes
3
answers
5k
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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 ...
8
votes
1
answer
255
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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
165
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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 ...
21
votes
2
answers
993
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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 \...
3
votes
2
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679
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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
433
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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 ...
9
votes
2
answers
833
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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 ...
14
votes
2
answers
1k
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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,\...
34
votes
2
answers
2k
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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 ...
16
votes
4
answers
3k
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Is there an infinite group with exactly two conjugacy classes?
Is there an infinite group with exactly two conjugacy classes?