Show that the group $(\Bbb Z,+)$ cannot be expressed as an internal direct product of two non-trivial subgroups.
My solution goes like this:
Any non-trivial subgroups of $(\Bbb Z,+)$ is of the form $n\Bbb Z$, $n\neq 0,1$. Let $h\Bbb Z,k\Bbb Z$ be two such non-trivial subgroups of $\Bbb Z$, where $h,k\in\Bbb Z$. So, $h\Bbb Z k\Bbb Z=\{hakb,\forall a,b\in\Bbb Z\}.$ Now, $\exists g\in\Bbb Z$, such that $\gcd(g,hk)=1$. Hence, $g\notin k\Bbb Z h\Bbb Z$. Hence, $\Bbb Z\neq k\Bbb Z h\Bbb Z$, where $k\Bbb Z, h\Bbb Z$ were arbitary non-trivial subgroup. Thus, $(\Bbb Z,+)$ cannot be expressed as an internal direct product of two non-trivial subgroups.
Is the above solution correct? If not, where is it going wrong?
Best Answer
You're confusing sums with products. Doing $h\mathbb{Z}k\mathbb{Z}$ is not the way and is meaningless in this context.
Since the operation is $+$, the condition for being a direct product of two subgroups would be $$ \mathbb{Z}=H+K,\quad H\cap K=\{0\} $$ where $H+K=\{a+b:a\in H,b\in K\}$.
Now, suppose $h\ge 0$ and $k\ge 0$. Can you have $$ h\mathbb{Z}\cap k\mathbb{Z}=\{0\} $$ without one among $h$ and $k$ being zero?