Question

Let G and H be groups. If $G\times H$ is a cyclic group, prove that G and H are both cyclic. (Exercise 12 shows that the converse is false)

Short answer

It has been proved G and H both are cyclic.

Step-by-step solution

Suppose the subset of G×H

Let us suppose $A=\left\{\right(g,e_H\left)\right|g\in G\}$is a subset of $G\times H$.

It is clear that A is a subgroup of$G\times H$

Since A is a subgroup and $G\times H$is cyclic therefore, A is also cyclic.

Prove that G is cyclic

If A is cyclic then (a, e_H) is a generator of this subgroup.

It implies that a is a generator of G.

Hence, G is cyclic.

On similar lines, H is also cyclic.

Conclusion

Hence, GandH both are cyclic.