Fibres of a group homomorphism are cosets of its kernel
= Fibres of a group homomorphism are cosets of its kernel
{title2=$\varphi(g)=\varphi(h)\iff h^{-1}g\in\ker\varphi$}
For a <group homomorphism> $\varphi:G\to H$, two inputs give the same output exactly when $\varphi(h^{-1}g)=e_H$. Consequently the <fibre> over an output $\varphi(h)$ is the left <coset> $h\ker\varphi$. These <cosets> identify precisely the inputs indistinguishable to $\varphi$, which is the mechanism behind the <first isomorphism theorem>.