Question

A group $\mathit{G}$ is said to be indecomposable if it is not the direct product of two of its proper normal subgroups. Prove that each of these groups is indecomposable:$\mathrm{\mathbb{Z}}$

Short answer

It has been proved that, $\mathbb{Z}$is an indecomposable group.

Step-by-step solution

Step-by-Step SolutionStep 1: Definition of Indecomposable

A group$\mathbf{G}$ is said to be indecomposable if it is not the direct product of its two proper normal subgroups.

Prove that ℤ is indecomposable

Any two non-zero subgroups of $\mathrm{\mathbb{Z}}$are cyclic.

So, the two subgroups coincide with $⟨m⟩$and $⟨n⟩$for some $m,n\in \mathbb{Z}$.

Hence, their intersection is non-trivial say $⟨k⟩$, where $k=lcm(m,n)$.

Therefore,$\mathrm{\mathbb{Z}}$ is not the direct product of these subgroups.

Thus, it can be concluded that$\mathrm{\mathbb{Z}}$ is indecomposable.