Let G be a group with exactly two nontrivial proper subgroups. Show that G is cyclic. What are the possible orders of G?
Assume G has no nontrivial subgroups.
Because G is trivial by the assumption above, then G is cyclic.
If G has exactly one nontrivial subgroups H, consider the subgroup generated by a nonidentity element g in G/H
Now suppose that H and K are the only nontrivial subgroups of G. Recall that a group is never the union of two proper subgroups....
(Above is the start I have to my proof. I haven;t finished proving it is a cyclic group and can't figure out how to find what the possible orders may be.)