# Tag Info

Accepted

• 14.2k
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 ...
• 1,079

### 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 ...
• 9,205

### 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 ...
• 37.7k
Accepted

### Are the character degrees determined by the conjugacy class sizes?

SmallGroup(128,227) and SmallGroup(128,731)) are counterexamples. ...
• 34.3k
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### 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 ...
• 61.6k
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### 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 ...
• 43.3k
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• 43.3k

### 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 &...
• 14.2k

### 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. ...
• 5,433
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### 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 ...
• 9,205

### 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 ...
• 2,653
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### 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 ...
• 43.3k
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 ...

### 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 ...
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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), ...