What does it mean by $|A_5/K|$ and how to prove that $|A_5/K|\geq 5$?

Question

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.

Answer 1

$G/H$ is standard notation for the set of left cosets of $H$ in $G$. See https://en.wikipedia.org/wiki/Coset#Notation for example. In particular, $|G/H|=|G:H|$, as you surmised.

(When $H$ happens to be normal, then this notation is also used for the natural group on this set, as you know.)

With the notation question out of the way, indeed you need $K<A_5$ for the question to be correct.