# quotient group

Before defining quotient groups^{}, some preliminary definitions must be introduced and a few established.

Given a group $G$ and a subgroup^{} $H$ of $G$, the http://planetmath.org/node/122relation^{} ${\sim}_{L}$ on $G$ defined by $a{\sim}_{L}b$ if and only
if ${b}^{-1}a\in H$ is called *left congruence ^{} modulo $H$*; similarly the relation defined by $a{\sim}_{R}b$ if
and only if $a{b}^{-1}\in H$ is called

*congruence modulo $H$*(observe that these two relations coincide if $G$ is abelian

^{}).

###### Proposition.

Left (resp. right) congruence modulo $H$ is an equivalence relation^{} on $G$.

###### Proof.

We will only give the proof for left congruence modulo $H$, as the for right congruence modulo $H$ is analogous.
Given $a\in G$, because $H$ is a subgroup, $H$ contains the identity^{} $e$ of $G$, so that ${a}^{-1}a=e\in H$; thus $a{\sim}_{L}a$, so ${\sim}_{L}$ is http://planetmath.org/node/1644reflexive^{}. If $b\in G$ satisfies $a{\sim}_{L}b$, so that ${b}^{-1}a\in H$, then by the of $H$ under the formation of inverses^{}, ${a}^{-1}b={({b}^{-1}a)}^{-1}\in H$, and $b{\sim}_{L}a$; thus ${\sim}_{L}$ is symmetric^{}. Finally, if $c\in G$, $a{\sim}_{L}b$, and $b{\sim}_{L}c$, then we have ${b}^{-1}a,{c}^{-1}b\in H$, and the closure of $H$ under the binary operation^{} of $G$ gives ${c}^{-1}a=({c}^{-1}b)({b}^{-1}a)\in H$, so that $a{\sim}_{L}c$, from which it follows that ${\sim}_{L}$ is http://planetmath.org/node/1669transitive^{}, hence an equivalence relation.
∎

It follows from the preceding that $G$ is partitioned into mutually disjoint, non-empty equivalence
classes^{} by left (resp. right) congruence modulo $H$, where $a,b\in G$ are in the same equivalence class if and only if $a{\sim}_{L}b$ (resp. $a{\sim}_{R}b$); focusing on left congruence modulo $H$, if we denote by $\overline{a}$ the equivalence class containing $a$ under ${\sim}_{L}$, we see that

$$\begin{array}{cc}\hfill \overline{a}& =\{b\in G\mid b{\sim}_{L}a\}\hfill \\ & =\{b\in G\mid {a}^{-1}b\in H\}\hfill \\ & =\{b\in G\mid b=ah\text{for some}h\in H\}=\{ah\mid h\in H\}\text{.}\hfill \end{array}$$ |

Thus the equivalence class under ${\sim}_{L}$ containing $a$ is simply the left coset^{} $aH$ of $H$ in $G$. Similarly the equivalence class under ${\sim}_{R}$ containing $a$ is the right coset $Ha$ of $H$ in $G$ (when the binary operation of $G$ is written additively, our notation for left and right cosets becomes $a+H=\{a+h\mid h\in H\}$ and $H+a=\{h+a\mid h\in H\}$). Observe that the equivalence class under either ${\sim}_{L}$ or ${\sim}_{R}$ containing $e$ is $eH=H$. The *index* of $H$ in $G$, denoted by $|G:H|$, is
the cardinality of the set $G/H$ (read “$G$ modulo $H$” or just “$G$ mod $H$”) of left cosets of $H$ in $G$ (in fact, one may demonstrate the existence of a bijection^{}
between the set of left cosets of $H$ in $G$ and the set of right cosets of $H$ in $G$, so that we may well take $|G:H|$ to be the cardinality of the set of right cosets of $H$ in $G$).

We now attempt to impose a group on $G/H$ by taking the of the left cosets containing the elements $a$ and $b$, respectively, to be the left coset containing the element $ab$; however, because this definition requires a choice of left coset representatives, there is no guarantee that it will yield a well-defined binary operation on $G/H$. For the of left coset to be well-defined, we must be sure that if ${a}^{\prime}H=aH$ and ${b}^{\prime}H=bH$, i.e., if ${a}^{\prime}\in aH$ and ${b}^{\prime}\in bH$, then ${a}^{\prime}{b}^{\prime}H=abH$, i.e., that ${a}^{\prime}{b}^{\prime}\in abH$. Precisely what must be required of the subgroup $H$ to ensure the of the above condition is the content of the following :

###### Proposition.

The rule $\mathrm{(}a\mathit{}H\mathrm{,}b\mathit{}H\mathrm{)}\mathrm{\mapsto}a\mathit{}b\mathit{}H$ gives a well-defined binary operation on $G\mathrm{/}H$ if and only if $H$ is a normal subgroup^{}
of $G$.

###### Proof.

Suppose first that of left cosets is well-defined by the given rule, i.e, that given ${a}^{\prime}\in aH$ and
${b}^{\prime}\in bH$, we have ${a}^{\prime}{b}^{\prime}H=abH$, and let $g\in G$ and $h\in H$. Putting $a=1$, ${a}^{\prime}=h$, and $b={b}^{\prime}={g}^{-1}$, our hypothesis^{} gives $h{g}^{-1}H=e{g}^{-1}H={g}^{-1}H$; this implies that $h{g}^{-1}\in {g}^{-1}H$, hence that $h{g}^{-1}={g}^{-1}{h}^{\prime}$ for some ${h}^{\prime}\in H$. on the left by $g$ gives $gh{g}^{-1}={h}^{\prime}\in H$, and because $g$ and $h$ were chosen arbitrarily, we may conclude that $gH{g}^{-1}\subseteq H$ for all $g\in G$, from which it follows that $H\u22b4G$. Conversely, suppose $H$ is normal in $G$ and let ${a}^{\prime}\in aH$ and ${b}^{\prime}\in bH$. There exist ${h}_{1},{h}_{2}\in H$ such that ${a}^{\prime}=a{h}_{1}$ and ${b}^{\prime}=b{h}_{1}$; now, we have

$${a}^{\prime}{b}^{\prime}=a{h}_{1}b{h}_{2}=a(b{b}^{-1}){h}_{1}b{h}_{2}=ab({b}^{-1}{h}_{1}b){h}_{2}\text{,}$$ |

and because ${b}^{-1}{h}_{1}b\in H$ by assumption^{}, we see that ${a}^{\prime}{b}^{\prime}=abh$, where $h=({b}^{-1}hb){h}_{2}\in H$
by the closure of $H$ under in $G$. Thus ${a}^{\prime}{b}^{\prime}\in abH$, and because left cosets
are either disjoint or equal, we may conclude that ${a}^{\prime}{b}^{\prime}H=abH$, so that multiplication^{}
of left cosets is indeed a well-defined binary operation on $G/H$.
∎

The set $G/H$, where $H$ is a normal subgroup of $G$, is readily seen to form a group under the well-defined
binary operation of left coset multiplication (the of each group follows from that of $G$), and is called a *quotient* or *factor group* (more specifically
the *quotient of $G$ by $H$*). We conclude with several examples of specific quotient groups.

###### Example.

A standard example of a quotient group is $\mathbb{Z}/n\mathbb{Z}$, the quotient of the of integers by the cyclic subgroup generated by $n\in {\mathbb{Z}}^{+}$; the order of $\mathbb{Z}/n\mathbb{Z}$ is $n$, and the distinct left cosets of the group are $n\mathbb{Z},1+n\mathbb{Z},\mathrm{\dots},(n-1)+n\mathbb{Z}$.

###### Example.

Although the group ${Q}_{8}$ is not abelian, each of its subgroups its normal, so any will suffice for the formation
of quotient groups; the quotient ${Q}_{8}/\u27e8-1\u27e9$, where $\u27e8-1\u27e9=\{1,-1\}$ is the cyclic subgroup of ${Q}_{8}$ generated by $-1$, is of order $4$, with elements $\u27e8-1\u27e9,i\u27e8-1\u27e9=\{i,-i\},k\u27e8-1\u27e9=\{k,-k\}$ , and $j\u27e8-1\u27e9=\{j,-j\}$. Since each non-identity element of ${Q}_{8}/\u27e8-1\u27e9$ is of order $2$, it is isomorphic^{} to the Klein $4$-group $V$. Because each of $\u27e8i\u27e9$, $\u27e8j\u27e9$, and $\u27e8k\u27e9$ has order $4$, the quotient of ${Q}_{8}$ by any of these subgroups is necessarily cyclic of order $2$.

###### Example.

The center of the dihedral group^{} ${D}_{6}$ of order $12$ (with http://planetmath.org/node/2182presentation^{} $\u27e8r,s\mid {r}^{6}={s}^{2}=1,{r}^{-1}s=sr\u27e9$) is $\u27e8{r}^{3}\u27e9=\{1,{r}^{3}\}$; the elements of the quotient ${D}_{6}/\u27e8{r}^{3}\u27e9$ are $\u27e8{r}^{3}\u27e9$, $r\u27e8{r}^{3}\u27e9=\{r,{r}^{4}\}$, ${r}^{2}\u27e8{r}^{3}\u27e9=\{{r}^{2},{r}^{5}\}$, $s\u27e8{r}^{3}\u27e9=\{s,s{r}^{3}\}$, $sr\u27e8{r}^{3}\u27e9=\{sr,s{r}^{4}\}$, and $s{r}^{2}\u27e8{r}^{3}\u27e9=\{s{r}^{2},s{r}^{5}\}$; because

$$s{r}^{2}\u27e8{r}^{3}\u27e9r\u27e8{r}^{3}\u27e9=s{r}^{3}\u27e8{r}^{3}\u27e9=s\u27e8{r}^{3}\u27e9\ne sr\u27e8{r}^{3}\u27e9=r\u27e8{r}^{3}\u27e9s{r}^{2}\u27e8{r}^{3}\u27e9\text{,}$$ |

${D}_{6}/\u27e8{r}^{3}\u27e9$ is non-abelian^{}, hence must be isomorphic to ${S}_{3}$.

Title | quotient group |

Canonical name | QuotientGroup |

Date of creation | 2013-03-22 12:04:06 |

Last modified on | 2013-03-22 12:04:06 |

Owner | azdbacks4234 (14155) |

Last modified by | azdbacks4234 (14155) |

Numerical id | 35 |

Author | azdbacks4234 (14155) |

Entry type | Definition |

Classification | msc 20-00 |

Synonym | factor group |

Synonym | quotient |

Related topic | Group |

Related topic | NormalSubgroup |

Related topic | Subgroup |

Related topic | EquivalenceRelation |

Related topic | Coset |

Related topic | NaturalProjection |

Defines | left congruence modulo a subgroup |

Defines | right congruence modulo a subgroup |

Defines | index |