Proving that a set of functions with the Dirichlet convolution operation is a group

Question

I want to prove that the set $S:(f:\Bbb Z\to\Bbb R);f(1)\neq 0)$ with the Dirichlet Convolution operator is a group.

My process:

1. $f,g:\Bbb Z\to\Bbb R$ and $f*g:\Bbb Z\to\Bbb R$ so $f*g\in S$

2. $f*(g*h)=(f*g)*h$ via properties of Dirichlet Convolution

3. $f*\delta_{1}=\delta_{1}*f=f$ where the neutral element is the Kronecker Delta of $1$.

4. $f*f^{-1}=\delta_{1}$ where $f^{-1}=\frac{-1}{f(1)}\sum_{d|n,d<n}f(\frac{n}{d})f^{-1}(d)$ follows from Dirichlet Inverses

Thus all four criteria for a group are satisfied and $S:(S,*)$ is a group. Furthermore, since $f*g=g*f$, $S$ is abelian.

Is my process correct? I feel as though the inverse condition is a little iffy so if anyone can help verify if what I did was valid that would be great.