A question about conjugate elements of group Let $G$ be a group. $a,b,x,y$ are elements of $G$.If $ab=ba$, $xy=yx$, $a$ is conjugate to $x$ and $b$ is conjugate to $y$ and $(o(a),o(b))=1$, then can we get $ab$ is conjugate to $xy$? I can’t give a contradictory example.

The question was basically answered in the comments, so I will expand them to an answer.
If $a$ is conjugate to $x$, say $a^g = x$, then $(ab)^g = a^g b^g = b^g a^g$. So by replacing $a$ with $a^g$, we may assume that $a =x$.
So the question is the following. Suppose that $xy = yx$ and $xy' = y'x$, where $y$ and $y'$ are conjugate in $G$, and $\gcd(o(x), (y)) = 1$. Must $xy$ and $xy'$ be conjugate in $G$?
Well, in this situation $xy$ and $xy'$ are conjugate if and only if $y,y'$ are conjugate in $C_G(x)$. (I leave the proof of this fact to the reader of this answer.)
So to find a counterexample, look for $y,y' \in C_G(x)$ which are conjugate in $G$ but not in $C_G(x)$. There are many such examples. You can find some in a suitable dihedral group $G$, take $x$ to be an element in the cyclic subgroup of index $2$ in $G$.
