# Union of conjugacy classes of a subgroup

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 $$pqr$$, $$p primes. Prove the $$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.

• @user1729 Edited. Jul 6, 2020 at 10:52
• I've retracted my close vote, and turned my downvote into an upvote :-) Jul 6, 2020 at 13:32
• I am sorry but I am reluctant to retract my close vote. The problem I have is that the answer to the question is clearly yes if $G$ is abelian and, more generally, if $H$ is a normal subgroup of $G$, because in these cases we have $S=H$. So we have to look at non-normal subgroups of groups, and then the smallest example shows that the answer to the question is no. Surely, before asking a question like this, you should at least check a few small examples? Jul 6, 2020 at 18:22
• @DerekHolt True. Earlier even I used to solve a problem by doing every single step myself. But this method takes a lot of time. So now, whenever I fear I'm heading the wrong way, I ask for help. In this case, while solving the original problem, I encountered the concept 'Union of conjugacy classes of a subgroup' which appeared unfamiliar, so I feared I'm on wrong path, so I didn't put much effort and asked for help. Jul 7, 2020 at 6:37
• OK, I have now retracted my close vote! Jul 7, 2020 at 7:34

Let $$G=S_3$$, $$H=C_2$$. Then $$|S|=4, |G|=6$$.