Let $H$ be a subgroup of the finite group $G$. For $h\in H$, let $C_h$ be the conjugacy class of $h$ in $G$. $S=\bigcup_{h\in H} C_h$.
Whether $\# S\mid |G|$?
Edit after @user1729 's comment:
I encountered this question while solving this question:
Herstein ch2.12 question 12a
Let $G$ be a group of order $pqr$, $p<q<r$ primes. Prove the $r$-Sylow subgroup is normal in $G$.
My attempt to the original question:
By third part of Sylow theorem, number of $r$-Sylow subgroups is $1$ or $pq$.
Suppose it is $pq$.
Fix an $r$-Sylow subgroup, say $H$. Since $r$-Sylow subgroups are conjugates and since they intersect trivially,
$\sum_{i}(C_{h_i})=pqr-pq+1$, where $h_i$ is representative from a conjugacy class.
If the answer to this post is 'yes', the above equation gives a contradiction and the original question is solved.
Best Answer
Let $G=S_3$, $H=C_2$. Then $|S|=4, |G|=6$.