For which $n$ is the group $\mathbb Z_n$ isomorphic to the subgroup of this group generated by $4$.

Question

The exercise is as follows:

1. Consider the group $\mathbb Z_{24}$.
   (a) For which $n$ is the group $\mathbb Z_n$ isomorphic to the subgroup of this group generated by the element $4$?

I'm a little confused by the wording. I understand that the element $4$ generates $$\{0,4,8,12,16, 20\}$$ but I am not sure how to relate this to the problem. If someone could show me how this relates to the question, or rephrase the question, it would be appreciated. Thanks.

Answer 1

The question is asking you to find one value of $n$ such that $\mathbb{Z}_n$ is isomorphic to the subgroup of $\mathbb{Z}_{24}$ generated by 4.

If all you need to do is find one such $n$, Noah's hint should suffice.

If you want to prove your choice of $n$ is correct, you will need to construct an isomorphism. Let $S$ be the subgroup generated by 4. You want to construct some $\phi : \mathbb{Z}_n \rightarrow S$ (you may switch the domain and codomain if you wish) such that:

• $\phi$ is a bijection
• For any $k_1$, $k_2 \in \mathbb{Z}_n$, we have $\phi(k_1 +_{n} k_2) = \phi(k_1) +_{24} \phi(k_2)$

where $+_{n}$ is addition modulo $n$. This will create an isomorphism between $\mathbb{Z}_n$ and $S$, thus showing they are isomorphic. Taking Noah's hint into account, you would just need to construct one such $\phi$.