Answer to Question #139772 in Abstract Algebra for J

Question

Prove that if (G,*) is a group, and if the only subgroups of G are G and {e}, then G is cyclic.

Answer 1

Let a be a nonidentity element of G. We consider the group H generated by single element a.

H=<a>. Clearly all elements of H must belong to G. Therefore H becomes a subgroup of G.

But as a$\neq e$, H$\neq \{e\}$ . Therefore H=<a>=G. (as the only subgroups of G are $\{e\}, G)$

Hence G becomes cyclic.