UA-AUG-AUMAT229 Quotients: Pre-read

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 $\equiv$ is an equivalence relation on those objects, write $A / \equiv$ to mean the collection of equivalence classes. This collection is called the quotient of $A$ modulo $\equiv .$

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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.

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Example 15.1.3.

Consider $Z$ with the equivalence relation of congruence modulo $4 .$ That is, we have

m \equiv_{4} n

if both $m$ and $n$ are the same amount greater than a multiple of $4 .$

The congruence classes are

\begin{aligned} [0] & = {\dots, - 4, 0, 4, 8, \dots}, \\ [1] & = {\dots, - 3, 1, 5, 9, \dots}, \\ [2] & = {\dots, - 2, 2, 6, 10, \dots}, \\ [3] & = {\dots, - 1, 3, 7, 11, \dots}, \end{aligned}

so the quotient is

(Z / \equiv_{4}) = {[0], [1], [2], [3]} .

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In the case of the partition of a group

G

into the (left) cosets of a subgroup

H,

we write

G / H

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to denote the collection of cosets.

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Example 15.1.4.

Consider $G = Z$ and $H = ⟨ 4 ⟩ .$ The (left) cosets of $H$ are

\begin{aligned} H = 0 + H & = {\dots, - 4, 0, 4, 8, \dots}, \\ 1 + H & = {\dots, - 3, 1, 5, 9, \dots}, \\ 2 + H & = {\dots, - 2, 2, 6, 10, \dots}, \\ 3 + H & = {\dots, - 1, 3, 7, 11, \dots}, \end{aligned}

so the quotient is

Z / ⟨ 4 ⟩ = {H, 1 + H, 2 + H, 3 + H} .

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Note 15.1.5.

You might notice that Example 15.1.3 and Example 15.1.4 look very similar …