"Let G be a nontrivial group with no proper subgroups except the trivial one. Show that G is finite and that the order of G is prime."
Okay, I think that maybe we know that G is finite because if it was infinite it you could always have an infinite subgroup that operations could not leave? I'm not sure how to prove that and I am totally stumped about how to prove that the order of G is prime.
I Know that if the order of G is even there must be an element
such that
where
is the identity element. Therefore you could have a subgroup with just
. But this only proves that finite group with no proper subgroups must have odd order, not necessary prime.
Any help would be appreciated.