Identify a group $G$ that has subgroups isomorphic to $\mathbb{Z}_n$ for all positive integer $n$

Question

Identify a group $G$ that has subgroups isomorphic to $\mathbb{Z}_n$ for all positive integer $n$.

Source: Gallian Contemporary Algebra, Isomorphism Chapter

I am stuck at the very premise of this question. It seems that the question is asking for a single group $G$ which would be isomorphic to every $\mathbb{Z}_n$ ever possible. However, how would I define a bijective mapping from $G$ to $\mathbb{Z}_n$ if their cardinality differs?

According to me, the question itself is flawed. Am I right? If not, what is the mistake in my understanding?

Answer 1

The question is not asking for the parent group to be isomorphic to all $\mathbb Z_n$, only that for each $n$ it has a subgroup isomorphic to $\mathbb Z_n$. Accordingly, the circle group under addition $\mathbb R/\mathbb Z$ fits the requirements with $\mathbb Z_n\cong\{k/n:k\in\mathbb Z,0\le k<n\}$.

Answer 2

Two easy examples:

• $\displaystyle G = \prod_{n \in \mathbb N} \mathbb{Z}_n$

• $G = \mathbb C^\times$