Is there a cyclic group such that is isomorphic to Z∗16?

Question

How do I find an additive group modulo $n$ $Z_n$ which is isomorphic to $Z^*_{16}$ (group with multiplication modulo 16)?

Answer 1

There are eight residue classes that are relatively prime to $16$, so one should only wonder whether the group of units of $\mathbb{Z}/16$ is isomorphic to the cyclic group $\mathbb{Z}/8$. But the four elements $1,7,9,15$ all have order two in the group of units, so it cannot possibly be cyclic.

Answer 2

There is no such cyclic group, since $(\mathbf Z/16\mathbf Z)^\times $ is isomorphic to Klein's Vierergruppe, i.e. it is the product of a cyclic group of order $2$ and a cyclic group of order $4$, since its elements are: $$\pm 1,\enspace \pm 3,\enspace \pm 5=\mp3^3,\enspace \pm 7=\mp3^2.$$

More generally, one shows $(\mathbf Z/2^r\mathbf Z)^\times \simeq \mathbf Z/2 \mathbf Z\times\mathbf Z/2^{r-2} \mathbf Z$, and $3$ generates a subgroup of order $2^{r-2}$.