I would like to prove the following statement.
Let $G$ be a group, $M$ is a maximal subgroup of $G$ and $M \trianglelefteq G$. Then $[G:M]$ is finite and equal to a prime.
This problem was already posted here several times (e.g. If a maximal subgroup is normal, it has prime index). However, all the proofs I found used the correspondence theorem (they are quite intuitive). Is there another nice way to prove the statement without the correspondence theorem?
As the correspondence theorem is not an easy one to prove (at least I think so), I would like to rather not introduce it just for this problem. Do you have any other ideas on how to tackle this problem?
Thank you very much for your help!