Problem: "Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime. (Do not assume at the outset that G is finite)." Basically, I'm pretty sure I can do this problem. I'm just unsure of how to prove that all infinite groups have some proper nontrivial subgroups. I also thought about just proving that all infinite groups and all groups with order not prime implies they have proper nontrivial subgroups. Either way I'm stuck on setting this up, any help would be of great help.