Show by left coset action that if $H$ is proper subgroup of $A_5$, then $[G:H]>4$

Question

Show that if $H$ is proper subgroup of $A_5$, then $[G:H]>4$ by considering the left coset action of $A_5$ on set of left cosets of $H$.

So I note that this is equivalent to saying $15 > |H| $ however I can't see a nice way of eliminating the few specific cases by group actions.

Also I would think orbit-stabiliser theorem is involved but I cannot see how to use it (and orbit stabiliser seems to imply Lagrange's theorem as a special case anyway).

Any hints?

Answer 1

A5 acts on the set of left cosets say there are n where n≤4. In this case there exists a group homomorphism from A5 to Sn. A5 is simple so this group homomorphism has to be one one. Thus A5 is isomorphic to a subgroup of Sn. but cardinality of A5 is 60 while any subgroup of Sn n≤4 has to have cardinality ≤24. This is a contradiction. Thus n>4