The quotient map by a finite normal subgroup is a quasi-isometry of word metric spaces

Given: The hypotheses of the Statement.

A group is finitely generated when some finite subset generates it (Finitely generated groups).

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

Word length is defined on every element and satisfies $\mathrm{\mid }gh{\mathrm{\mid }}_S\le \mathrm{\mid }g{\mathrm{\mid }}_S+\mathrm{\mid }h{\mathrm{\mid }}_S$, $\mathrm{\mid }{g}^{-1}{\mathrm{\mid }}_S=\mathrm{\mid }g{\mathrm{\mid }}_S$, and $\mathrm{\mid }g{\mathrm{\mid }}_S=0$ exactly when $g$ is the identity (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

A map is $(L,C)$-coarse Lipschitz when $d(f(x),f(x^{\mathrm{\prime }}))\le Ld(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding when in addition $L^{-1}d(x,x^{\mathrm{\prime }})-C\le d(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

The quotient group, or factor group, $G\mathrm{/}N$ has the left cosets (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$).

The subgroup $N$ is normal in $G$ when (Normal subgroup: invariance under conjugation).

A set $A$ is finite when $A\approx n$ for some $n\in \mathbb{N}$. (The cardinality $\mid A\mid$ of a finite set).