abstract algebra – Does an Abelian group under multiplication modulo $n$ contain $0$?

I read the following definition in a notebook.

The set $mathbb{Z}^{*}_{n}$ that consists of all integers $i = 0, 1, …, n-1$ for which the $operatorname{gcd}(i, n) = 1$ forms an abelian group under the binary operation multiplicaiton modulo $n$. The identity element, of course, is $1$.

But I think it should be,

The set $mathbb{Z}^{*}_{n}$ that consists of all integers $i = 1, …, n-1$ for which the $operatorname{gcd}(i, n) = 1$ forms an abelian group under the binary operation multiplicaiton modulo $n$. The identity element, of course, is $1$.

Or could it be that both the definition are valid?