Question

Let N be a cyclic normal subgroup of a group G , and H any subgroup of N . Prove that H is a normal subgroup of G .[Compare Exercise 14]

Short answer

It has been proved that H is a normal subgroup of G .

Step-by-step solution

Given information

It is given that N is a cyclic normal subgroup of G .

Thus,$aN{a}^{-1}=N$ for every $a\in G$.

Also, H is a subgroup of N .

Since N is cyclic therefore H is also cyclic and a subgroup of G.

Prove that H  is normal in  G.

Let $h\in H$and $a\in G$.

Consider $aH{a}^{-1}$ ; it has the same order as of H .

Since cyclic subgroups of the same order are isomorphic,

Therefore,

Since N is cyclic, therefore, it is generated by $\left(n\right)$.

Since $\left(n\right)$ is normal and $h\in \left(n\right)$,

Thus $aH{a}^{-1}\in \left(n\right)$

So $aH{a}^{-1}=H$

Conclusion

Hence, H is a normal subgroup of G .