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What are the reasons for the name "Quotient" in Quotient map, group - etc? Overhead picture shows each of the three cosets in $A_4$ is mapped to a single - gray - node. But this isn't division?

This is from Nathan Carter page 169 Visual Group Theory.

Another picture from Nathan Carter page 274 based on page 182 exercise 8.12.
$G$ is a group, $H$ is a normal subgroup.
$q : G \to \frac{G}{H}$ is a quotient map. $\phi: H \to G$ is an embedding.
$\theta$ is a map into $H$ would satisfying the equation $Im(\theta) = \ker\phi$ with the smallest possible domain. $\theta'$ is a map from $\frac{G}{H}$ satisfying the equation $Im(q) = \ker(\theta')$ with the smallest possible domain.

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This probably comes from the fact that, if $H\subset G$ are finite groups


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