Must $G$ be cyclic?

Question

If $G$ is a group and $H_1,H_2$ are cyclic subgroups of $G$ with $G=H_1H_2=H_2H_1$ then must $G$ be cyclic?

Please help me !

Answer 1

Note that if $H_1$ and $H_2$ are subgroups of $G$, then $H_1H_2$ is a subgroup if and only if $H_1H_2 = H_2H_1$. So the assumption $G = H_1H_2 = H_2H_1$ is equivalent to $G = H_1H_2$.

One special case where $G = HK$ is when $G$ is the direct product of $H$ and $K$.

Is the direct product of two cyclic groups always cyclic?

Answer 2

Consider $G = D_4 = \{1,x,x^2,x^3,y,xy,x^2y,x^3y\}$ and set $H_1 = \{1,y\}$ and $H_2 = \{1,x,x^2,x^3\}$. Then $H_2H_1= G$ and furthermore

$$H_1H_2 = \{1,y,x,yx,x^2,yx^2,x^3,yx^3\} = G.$$

But obviously $G$ is not cyclic.

Answer 3

The group $\mathbb{Z}_2$ is cyclic but $\mathbb{Z}_2\oplus\mathbb{Z}_2$ is not cyclic.