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Section 15.1 Quotients: Pre-read

Recall that an equivalence relation partitions a collection into equivalence classes.

Definition 15.1.1. Quotient collection.

If \(A\) is a collection of objects and \(\mathord{\equiv}\) is an equivalence relation on those objects, write \(A/\mathord{\equiv}\) to mean the collection of equivalence classes. This collection is called the quotient of \(A\) modulo \(\mathord{\equiv}\text{.}\)

Warning 15.1.2.

A quotient is a collection of collections. That is, an object in the quotient is a collection of objects from the original collection.

Example 15.1.3.

Consider \(\Z\) with the equivalence relation of congruence modulo \(4\text{.}\) That is, we have

\begin{equation*} m \equiv_4 n \end{equation*}

if both \(m\) and \(n\) are the same amount greater than a multiple of \(4\text{.}\)

The congruence classes are

\begin{align*} \eqclass{0} \amp = \{ \dotsc, -4, 0, 4, 8, \dotsc \} \text{,} \\ \eqclass{1} \amp = \{ \dotsc, -3, 1, 5, 9, \dotsc \} \text{,} \\ \eqclass{2} \amp = \{ \dotsc, -2, 2, 6, 10, \dotsc \} \text{,} \\ \eqclass{3} \amp = \{ \dotsc, -1, 3, 7, 11, \dotsc \} \text{,} \end{align*}

so the quotient is

\begin{equation*} (\Z/\mathord{\equiv}_4) = \{ \eqclass{0}, \eqclass{1}, \eqclass{2}, \eqclass{3} \} \text{.} \end{equation*}

In the case of the partition of a group \(G\) into the (left) cosets of a subgroup \(H\text{,}\) we write

\begin{equation*} G/H \end{equation*}

to denote the collection of cosets.

Example 15.1.4.

Consider \(G = \Z\) and \(H = \gen{4}\text{.}\) The (left) cosets of \(H\) are

\begin{align*} H = 0 + H \amp = \{ \dotsc, -4, 0, 4, 8, \dotsc \} \text{,} \\ 1 + H \amp = \{ \dotsc, -3, 1, 5, 9, \dotsc \} \text{,} \\ 2 + H \amp = \{ \dotsc, -2, 2, 6, 10, \dotsc \} \text{,} \\ 3 + H \amp = \{ \dotsc, -1, 3, 7, 11, \dotsc \} \text{,} \end{align*}

so the quotient is

\begin{equation*} \Z/\gen{4} = \{ H, 1 + H, 2 + H, 3 + H \} \text{.} \end{equation*}