8.3_24E - Algebra 1

Question

If G is a cyclic group, prove that G/N is cyclic, where N is any subgroup of G.

Step-by-Step Solution

Answer

It is proved that $\mathit{G}\mathbf{/}\mathit{N}$is cyclic.

Step 1: Given information

• G is a cyclic group.
• N is any subgroup of G.

Step 2: Proving that is cyclic

Suppose that G is cyclic with generator g.

 

So, 

 

Now, consider

 

Taking   for some . So, there exists such that . We get,

 

 

It implies that

As we had considered that , so it must be a cyclic group with generator. Hence, it is proved that is cyclic.