What does it mean to "Decide to which group $G$ is isomorphic" for a given group $G$? I have a homework question which is 

Decide to which group $(\mathbb{Z}_n^*,\,\cdot\,)$ is isomorphic (classification of finite abelian groups), for
    (i) $n = 9$,
    (ii) $n = 15$. 

But I don't understand what it means. Should I decide whether $\mathbb{Z}_9^*$ and $\mathbb{Z}_{15}^*$ is isomorphic or I need to do something else?
 A: I'm assuming you mean $\mathbb Z_n^*$. 
Your question means that for 
$(i)$ Find the group that is isomorphic to $\mathbb Z_9^*$: you'll want to determine how to represent it as the decomposition into the direct product of cyclic groups (one or more factors): Express this isomorphism as the decomposition.
$(ii)$ Find the group that is isomorphic to $\mathbb Z_{15}^*$. Again, use decomposition to express this as the direct product of cyclic groups, and express this isomorphism as its decomposition.
For any group denoted by $\mathbb Z_n^*,\,$ alternatively denoted by $(\mathbb Z/n\mathbb Z, \cdot),\,$ there are $\phi(n)$ elements in this group, the elements being those integers (each representing a congruence class) strictly between $0$ and $n$ and which are coprime to $n$. $\;\phi(n) = n − 1\,$ if and only if $n$ is itself a prime. Once you determine the order of each group, decomposition into the direct product of cyclic products of the same order should easily fall out. (If you have further questions, feel free to comment below.)
