I saw the following exercise just now; it should have something to do with group actions:
Let $K\leq A_5$. Show that $|A_5/K|> 4$.
I discovered I don't really understand what this means. First of all, $A_5$ is simple, so $K$ is not normal unless $K=A_5$ or $K=1$. So the quotient set $A_5/K$ is NOT a group. I guess that $|A_5/K|=|A_5: K|=|A_5|/|K|$, but this doesn't seem to be right. Also, when $A_5=K$, $|A_5|/|K|=1$, so this is impossible.
What does this question mean? I am pretty shocked to be unable to understand it.