Question

If $\mathbf{p}$is a prime and $\mathbf{n}$is a positive integer, prove that ${\mathbf{\mathbb{Z}}}_{{\mathbf{p}}^{\mathbf{n}}}$is indecomposable.

Short answer

Answer:

It has been proved that, ${\mathrm{\mathbb{Z}}}_{p^n}$is an indecomposable group.

Step-by-step solution

Definition of Indecomposable

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

Prove that ℤpn is indecomposable

The non-zero subgroups of ${\mathrm{\mathbb{Z}}}_{p^n}$are $<1>,<p>,<p^2>,....<p^{n-1}>$.

Any of these two subgroups coincide non-trivially.

So ${\mathrm{\mathbb{Z}}}_{p^n}$cannot be the direct product of its subgroups.

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