# Algebraic structure on conjugacy classes

informally speaking, what algebraic structure does the set of conjugacy classes of a group carry?

Formally, I'm interested in natural operations on conjugacy classes. Let $$\mathsf{Grp}$$ be the category of groups and $$\mathsf{CC} : \mathsf{Grp} \to \mathsf{Set}$$ the functor mapping every group to its set of conjugacy classes. Then a natural operation of arity $$n \in \mathbb{N}$$ is a natural transformations of the form $$\mathsf{CC}^{\times n} \longrightarrow \mathsf{CC}.$$

Here are the ones that I know of (hat tip to YCor):

• Projecting onto one of the factors.
• In arity $$n = 1$$, taking a power $$x \mapsto x^k$$ for fixed $$k \in \mathbb{Z}$$, which descends to conjugacy classes.
• Composing the first kind with the second.
• In arity $$n = 0$$, returning the class of the neutral element.

Are there any other ones? In particular, are there nontrivial operations of arity $$> 1$$?

• Arity 1: for each $d$, the map $x\mapsto x^d$ factors modulo conjugation.
– YCor
Commented May 25 at 11:46
• @YCor, d'oh, of course...! I'll add that to the question. Commented May 25 at 11:47
• For power maps, you don’t need $k\in\mathbb{N}$. Negative $k$ works as well. Commented May 25 at 12:03
• thanks @JeremyRickard, corrected. Commented May 25 at 12:05
• Given a group $G$, there are also left/right $G$-actions \begin{align*}\lambda_G&\colon\mathrm{Z}(G)\times\mathrm{CC}(G)\to\mathrm{CC}(G),\\\rho_G&\colon\mathrm{CC}(G)\times\mathrm{Z}(G)\to\mathrm{CC}(G)\end{align*} given by left/right multiplication. It would be interesting to explore how the natural operations $\mathrm{CC}^{\times n}\to\mathrm{CC}$ interact with these, e.g. which ones would be maps of left/right $G$-sets componentwise? Alternatively, we could also study natural operations $\mathrm{CC}^{\boxtimes n}\to\mathrm{CC}$, where $\boxtimes$ is the tensor product of $G$-sets. Commented May 25 at 15:19

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. Write $$[a]$$ for the conjugacy class of $$a$$. Then, taking $$G$$ to be free on generators $$x$$ and $$y$$ and choosing a representative for the class $$f([x],[y])$$, i.e. a word $$w(x,y)$$ such that $$[w(x,y)]=f([x],[y])$$, by naturality we have $$[w(a,b)]=f([a],[b])$$ in general. In the free group on generators $$x,y,z$$ the words $$w(zxz^{-1},y)$$ and $$w(x,y)$$ are then conjugate. It is clear then that in the (reduced) word $$w(x,y)$$ either the letter $$x$$ does not occur or the letter $$y$$ does not.
Indeed, unless $$w$$ is of the form $$x^k$$ or $$y^k$$, by replacing $$w$$ with a conjugate we may assume $$w(x,y) = x^{a_1} y^{b_1} \cdots x^{a_n} y^{b_n}$$ where $$a_1, b_1, \dots, a_n, b_n$$ are all nonzero. Then $$w(zxz^{-1}, y) = z x^{a_1} z^{-1} y^{b_1} \cdots z x^{a_n} z^{-1} y^{b_n}$$. This expression is cyclically reduced and contains a $$z$$, so $$w(zxz^{-1}, y)$$ is not conjugate to $$w(x, y)$$.
• Not so fast. $w$ could be $x^k$ or $y^k$. Commented May 25 at 18:44