Question

If G is a cyclic group, prove that$\mathit{G}\mathbf{/}\mathit{N}$ is cyclic, where N is any subgroup of G.

Short answer

It is proved that $G/N$is cyclic.

Step-by-step solution

As given in the question

• Gis a cyclic group.
• Nis any subgroup of G.

Proving that G/N is cyclic

Suppose G is cyclic with generator g.

So,$G=⟨g⟩$

Now, consider$G/N=⟨Ng⟩$

Taking $Nx\in G/N$, for some $x\in G$. So, there exists$n\in \mathbb{Z}$such that $x=g^n$. We get

$\begin{array}{c}Nx=N{g}^n\\ =\underset{nterms}{\left(Ng\right).\left(Ng\right)\mathrm{.......}\left(Ng\right)}\\ ={\left(Ng\right)}^n\end{array}$

Which implies${\left(Ng\right)}^n\in ⟨Ng⟩$

As we had considered that$G/N=⟨Ng⟩$ , so it must be a cyclic group with generator $Ng$. Hence, it is proved that$G/N$ is cyclic.