17
votes
Accepted
Algebraic structure on conjugacy classes
I claim that there are no natural binary operations (and the same method should apply to any arity $\ge 2$) except those which depend on at most one of the two inputs:
Suppose that $f$ is natural. ...
- 55.9k
14
votes
Conjugacy classes as left Kan extension of forgetful functor
$\newcommand{\Gp}{\mathbf{Grp}}
\newcommand{\conj}{^{\mathbf{conj}}}
\newcommand{\Gpd}{\mathbf{Gpd}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\ho}{\mathrm{ho}}
\newcommand{\Fun}{\mathrm{Fun}}
\...
- 15.8k
11
votes
Accepted
Groups with three conjugacy classes that define an ordering
There are currently no known examples of bi-orderable groups where all positive elements are conjugate. The question of their existence appears as Problem 3.31 of the 2009 problem list Unsolved ...
- 1,089
11
votes
Conjugacy classes in towers of groups
YCor beat me to it, but I will post my answer anyway because it is rather different. I will construct a counterexample to the first question with $\Gamma = F_2 = F\{x,y\}$, the free group on two ...
- 9,617
10
votes
Center of a monoid ring
This was originally two questions, one asking about the center of a group ring and one asking about the center of a monoid ring. The answer for groups is quite simple but the answer for monoids is ...
- 38.6k
10
votes
Accepted
Are the character degrees determined by the conjugacy class sizes?
SmallGroup(128,227) and SmallGroup(128,731)) are counterexamples.
...
- 35.2k
10
votes
Accepted
Conjugacy classes in towers of groups
Here is a counterexample, with the Heisenberg group. Define by induction $a_0=1$ and $a_{n+1}=(a_n^2+1)a_n$. Clearly $a_n$ tends to infinity.
Let $\Gamma$ be the Heisenberg group, consisting of ...
- 63.9k
9
votes
Accepted
Why would dim primitive irrep divide size of some conjugacy class ?
I have not noticed this question before, though it was posted several years ago. As a comment on the question as a whole, and especially Question 1 asked in the text, there are likely to be many such ...
- 44.4k
9
votes
Accepted
A pair of non-conjugate subgroups: a simple proof
I think that the elements $g = \dfrac1{\sqrt2}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}^{\oplus3}$ and $h = \dfrac1 2\begin{pmatrix} 1 & 1 & 1 & 0 & 1 & 0 \\ -1 & 1 ...
- 12.9k
8
votes
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?
Oh, I did not know about this ongoing discussion on math overflow. Amri pointed out to me about this discussion today morning only. As I discussed with you in a private communication, I don't know how ...
- 899
8
votes
Accepted
Size of conjugacy classes in infinite groups
The answer is no: there exists a 2-generated group, having finite conjugacy classes of unbounded size.
Indeed B.H. Neumann (1937) produced a 2-generated group $G$ with normal subgroups $(H_n)_{n\ge 5}$...
- 63.9k
8
votes
Accepted
Number of conjugacy classes of pairs of commuting elements II
Revised answer, to cover more of question: The proof that $r_{G} \geq \frac{8p}{3}$ if $G$ is non-Abelian is tedious, but not too difficult, I think, and this is attained, (as you say),for $p = 3$ by ...
- 44.4k
7
votes
Conjugacy classes as left Kan extension of forgetful functor
An alternative to Maxime's sophisticated proof is to observe that $X$ and $L$ are both corepresented by the free group, because a group homomorphism from a free group (up to conjugacy) is the same as ...
- 409
7
votes
Accepted
Does $\mathrm{SL}_{n}(\mathbb{Z}/p^{2})$ have the same number of conjugacy classes as $\mathrm{SL}_{n}(\mathbb{F}_{p}[t]/t^{2})$?
I would have preferred to not answer my own question, but here it goes. Yes, the two groups have the same number of conjugacy classes and in fact, the groups $\mathrm{SL}_{n}(W_{2}(\mathbb{F}_{q}))$ ...
- 3,813
7
votes
Accepted
Constructing the largest finite group with a fixed number of conjugacy classes
Following @verret comment, here are the list of the largest finite groups $G_k$ with a fixed class number $k\le 14$ (i.e. the number of conjugacy classes of elements, or the number of irreducible ...
6
votes
Are the character degrees determined by the conjugacy class sizes?
Here is a general comment related to my answer to a previous MO question. If $\chi$ is a complex irreducible character of a finite group $G$, and $\chi$ takes a root of unity value at $x \in G$, then $...
- 44.4k
6
votes
What are the conjugacy classes of the category of ($\kappa$-small) sets?
So for endonorphisms up to isomorphisms, you're asking for a description of endomorphisms of sets. It sort of depends what kind of description you're looking for, but you could imagine something like &...
- 15.8k
6
votes
Accepted
Variety of conjugacy classes
As already noted, the quotient need not be even $T_1$. For example, $SL_2(\mathbb{C})/SL_2(\mathbb{C})$ is homeomorphic to $\mathbb{C}$ with double points at $\pm 2$. This quotient is not $T_1$ ...
- 8,529
6
votes
Size of conjugacy classes in infinite groups
The size of all conjugacy class of a group $G$ is bounded if and only if the derived subgroup $G^\prime$ of $G$ is finite.
This is a celebrated theorem of Neumann proved in the following paper:
B.H. ...
- 5,878
5
votes
Accepted
A probability problem in the conjugacy classes of symmetric group
Let $kp$ be the size of the support of $\sigma$. Let $1,2,3,4$ be four points of the ground set. The probability that $\sigma_1$ maps $1 \mapsto 2$ is $kp/n(n-1)$, because there is a $kp/n$ chance ...
- 9,617
5
votes
Number of conjugacy classes of a semi-direct product of two finite groups
For an example with equality, consider the order-$16$ central product of $D_4$ and $C_4$, i.e., the group (of order $16$) $G=NK$ with $N\cong D_4$, $K\cong C_4$, $|N\cap K|=2$ and $C_G(N)=K$. This can ...
- 2,959
5
votes
Accepted
Number of conjugacy classes of pairs of commuting elements
I think it is false in general that $r_{G} \geq p^{\frac{3}{2}},$ where $p$ is the largest prime divisor of the order of $G$.
If we take a Frobenius group $G$ of order $pq,$ where $p,q$ are primes ...
- 44.4k
5
votes
Accepted
Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically?
This question will likely be closed, but I'll try to give the OP some references before it is. First, let me answer the OP's question in the comments, regarding why this is vague and why the question ...
- 30.3k
4
votes
Maximum conjugacy class size in $S_n$ with fixed number of cycles
Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?
I will write $\prod_{i \geq 1}i^{a_i}...
- 30.1k
4
votes
Variety of conjugacy classes
What you look it at is exactly the quotient of the representation variety $$Hom({\mathbf Z},G)$$ of representations from the integers ${\mathbf Z}$ to $G$, by the $G$-action via conjugation.
The ...
- 10.4k
4
votes
Groups with three conjugacy classes that define an ordering
I have proposed a positive solution to this problem in a preprint entitled Hyperexponentially closed fields, to be found here, more precisely in Sections 10.1 and 10.2.
The solution is based on work ...
- 2,519
4
votes
Number of commuting pairs in p-group
Let me try to write an answer. As I said in comments, Higman's conjecture seems still to be open, and does not, in any case, predict a precise formula for $k(U)$.
As for the other question, as ...
- 44.4k
4
votes
Accepted
Conjugacy classes in the automorphism group of a simple Lie algebra
If $\mathfrak{s}$ is $K$-anisotropic, where $K$ is a real or $p$-adic field (this is equivalent to $\mathfrak{s}$ not containing $\mathfrak{sl}_2$, and also to the corresponding group be compact), ...
- 63.9k
4
votes
Fusing conjugacy classes II
No. Take $G={\rm SL}(2,{\Bbb R})$, $\ H=\{\,h(\lambda)={\rm diag}(\lambda, \lambda^{-1})\ |\ \lambda\in {\Bbb R}, \lambda>0\,\}$,
$$U=\bigg\{ u(a)=
\begin{pmatrix}
1 &a\\ 0&1
\end{pmatrix}\...
- 14.1k
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