I need to prove that if $G$ is an infinite group in which every non trivial proper subgroup is maximal, $G$ is simple.
I've found this: Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group. but it says nothing about maximals. I must show that there can't be normal proper subgroups, right? Well, if every proper non trivial subgroup is maximal, it means that there can't be another proper one that contains this and it's different. In other words: $H$ is maximal if there is no other proper subgroup $K$ such that $H\subset K$ strictly. I cannot see a connection to normality of such subgroups. Which should be the way to tackle this problem?