Question on a product of conjugacy classes

Question

Question:

Suppose $A$ is a conjugacy class of a group $G$ and $B$ is a conjugacy class of a group $H$. Can we claim, in general, that $A\times B$ is a conjugacy class of $G\times H$?

I've played around with some examples, and this has held, but I can't seem to get anywhere by proving it in general. Moreover, I haven't played around with infinite groups, but it "feels" like it should hold there too. Any help is greatly appreciated!

Answer 1

If $a\in A$, $b\in B$, $g\in G$ and $h\in H$ then $(a^g,b^h)$ is conjugate to $(a,b)$. Thus $A\times B$ is contained in the conjugacy class of $(a,b)$. On the other hand, if $(c,d)$ is conjugate to $(a,b)$ then there exists $(g,h)\in G\times H$ such that $(a,b)^{(g,h)}=(c,d)$. But the first term is just $(a^g,b^h)$, so already lies in $A\times B$.

Answer 2

Let $(g,h)\in G\times H$. Consider $(a,b)\in A\times B$. We have

$$\begin{align} (g,h)(a,b)(g,h)^{-1}&=(g,h)(a,b)(g^{-1},h^{-1})\\ &=(gag^{-1},hbh^{-1}). \end{align}$$

Also $gag^{-1}\in A$ and $hbh^{-1}\in B$. Thus $(g,h)(a,b)(g,h)^{-1}\in A\times B$.