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?
Best Answer
Both are valid. Note that $\operatorname{gcd}(0,n)=n\ne1$ (for $n>1$; but this is not serious restriction). Hence $0$ is excluded anyways.