Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups?

Question

Is there any External Direct Product which is cyclic but it is the product of two non-cyclic groups?

I know that for an EDP, say $G\times G'$ to be cyclic the $\gcd(o(G),o(G'))=1$.
But I am unable to find any EDP for the above problem.

Thanks in advance.

Answer 1

Suppose $H,K$ are not cyclic but $G=H\times K$ is. Then

$$H'=\{(h,e_K)\mid h\in H\}$$

is a noncyclic subgroup of $G$. But every subgroup of a cyclic group is itself cyclic!

Some Theorems you need:

• If $G_1,G_2$ are groups such that $G_1\cong G_2$ and $G_1$ is cyclic, then $G_2$ is cyclic. (Use the contrapositive of this.)
• Subgroups of a group are themselves groups.
• Here $H'\cong H$.