# 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?