Questions tagged [conjugacy-classes]
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20
votes
1answer
469 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 ...
1
vote
0answers
50 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? ...
0
votes
0answers
72 views
Complex symmetric matrices in a conjugacy class of SU(N)
Let $a$ be the following matrix in SU(N)
$$
A=\begin{bmatrix}a & \\ & a\\ &&\ddots \\&&&b\\&&&& b\\&&&&&\ddots\end{bmatrix}
$$
with $m$...
0
votes
0answers
89 views
On conjugacy classes of $M_n(F)$ where $F$ is a finite field
I want to determine all the orbits of $GL_n(F)$ acting on $Mat_n(F)$. But this seems too complicated, so I want to start with $Mat_3(F).$ But still, I have no idea. I know that Fulman has introduced a ...
2
votes
1answer
220 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 ...
11
votes
2answers
361 views
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)=\...
26
votes
3answers
1k views
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
1answer
231 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 $g_1^G\dotsb g_n^G$ ...
1
vote
0answers
135 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 ...
19
votes
2answers
703 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 \...
1
vote
2answers
421 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)}/{\...
10
votes
2answers
325 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 ...
8
votes
2answers
557 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 ...
12
votes
2answers
734 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,\...
30
votes
2answers
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 ...
13
votes
4answers
2k views
Is there an infinite group with exactly two conjugacy classes?
Is there an infinite group with exactly two conjugacy classes?
7
votes
3answers
556 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 ...
5
votes
4answers
1k 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 ...
8
votes
0answers
314 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 ...
