I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$
Is there any reference for this? Why this is true?
I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$. $Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$ Is there any reference for this? Why this is true? 


See Milne's online course notes on Étale Cohomology, Example 11.3, or Lei Fu's Étale Cohomology Theory, Proposition 5.7.20. (By passing to the direct limit over all $n$, you can even prove it for $\mathbf{Q}/\mathbf{Z}$.) You only need $X$ to be connected Noetherian. I am interested in alternative proofs not using torsors and Cech cohomology. BTW, for $X$ normal, it is also true for $\mathbf{Z}$ or $\mathbf{Q}$coefficientsboth sides are $0$ in this case. 

