Apologies with formatting. Nearly all the written n's are meant to be subscripts, but I can't quite figure that out yet
Let n ∈ $ℤ_n$ with n>1. Let $[a]_n$ ∈ $ℤ_n$ with $[a]_n$ not equal to $[0]_n$ and define a function
f:$ℤ_n$→$ℤ_n$ by f:$[b]_n$↦$[b]_n$*$[a]_n$
Prove that $[a]_n$ is a unit if and only if f is surjective and show $[a]_n$ is a unit if and only if f is injective.
I understand what it means to be a unit. I'm very confused as to how a function being surjective or injective will cause the element to be a unit itself.
Hint:
As this is a map from a finite set to itself, $f$ injective $\iff f$ surjective ($\iff f$ bijective).
On the other hand, $[a_n]$ is a unit iff and only if $[1]_n$ is attained.