Show that a group G has no nontrivial subgroups if and only if it is a cyclic group of prime order
If we're done. So, assume not. Then, let we must clearly have that but it isn't trivial and so it must be improper. Thus, . Now, if then by 's cyclicness we must have that there exists some such that . Since cannot be nontrivial or proper it follows that . The conclusion follows.