I used the code below hoping that the second set of equations aligned with & on each line would align but it doesn't seem to. It creates a space between the note and the equation on the same line but doesn't align the notes with each other.
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Let $f: G \rightarrow G'$ be a homomorphism with trivial kernel. Then $f$ is injective.
\begin{align}
f(a)&=f(b) \nonumber\\
f(a)^{-1} f(a) &= f(a^{-1})f(b) &(f(a)^{-1} = f(a^{-1})) \nonumber\\
e' &= f(a^{-1}b) &(f(ab) = f(a)f(b)) \nonumber\\
e &= a^{-1}b &(f^{-1}(e) = {e}) \nonumber
\end{align}