PlanetMath: quotient group

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

Given a group

and a subgroup

, the relation $\sim_L$ on

defined by $a\sim_L b$ if and only if $b^{-1}a\in H$ is called left congruence modulo

; similarly the relation defined by $a\sim_R b$ if and only if $ab^{-1}\in H$ is called right congruence modulo

(observe that these two relations coincide if

is abelian).

Proof. We will only give the proof for left congruence modulo

, as the argument for right congruence modulo

is analogous. Given $a\in G$ , because

is a subgroup,

contains the identity

, so that $a^{-1}a=e\in H$ ; thus $a\sim_L a$ , so $\sim_L$ is reflexive. If $b\in G$ satisfies $a\sim_L b$ , so that $b^{-1}a\in H$ , then by the closure of

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

under the binary operation of

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 transitive, hence an equivalence relation. $\qedsymbol$

It follows from the preceding proposition that

is partitioned into mutually disjoint, non-empty equivalence classes by left (resp. right) congruence modulo

, 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

, if we denote by $\bar{a}$ the equivalence class containing

under $\sim_L$ , we see that

Thus the equivalence class under $\sim_L$ containing

is simply the left coset

. Similarly the equivalence class under $\sim_R$ containing

is the right coset

(when the binary operation of

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

. The index of

, denoted by $\vert G:H\vert$ , is the cardinality of the set

(read “

modulo

" or just “

mod

") of left cosets of

(in fact, one may demonstrate the existence of a bijection between the set of left cosets of

and the set of right cosets of

, so that we may well take $\vert G:H\vert$ to be the cardinality of the set of right cosets of

We now attempt to impose a group on

by taking the product of the left cosets containing the elements

and

, respectively, to be the left coset containing the element

; 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

. For the operation of left coset multiplication 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

to ensure the satisfaction of the above condition is the content of the following proposition:

Proof. Suppose first that multiplication 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

, $a^\prime=h$ , and $b=b^\prime=g^{-1}$ , our hypothesis gives $hg^{-1}H=eg^{-1}H=g^{-1}H$ ; this implies that $hg^{-1}\in g^{-1}H$ , hence that $hg^{-1}=g^{-1}h^\prime$ for some $h^\prime\in H$ . Multiplication on the left by

gives $ghg^{-1}=h^\prime\in H$ , and because

and

were chosen arbitrarily, we may conclude that $gHg^{-1}\subseteq H$ for all $g\in G$ , from which it follows that $H\unlhd G$ . Conversely, suppose

is normal in

and let $a^\prime\in aH$ and $b^\prime\in bH$ . There exist $h_1,h_2\in H$ such that $a^\prime=ah_1$ and $b^\prime=bh_1$ ; now, we have

$\displaystyle a^\prime b^\prime=ah_1bh_2=a(bb^{-1})h_1bh_2 =ab(b^{-1}h_1b)h_2$ ,

and because $b^{-1}h_1b\in H$ by assumption, we see that $a^\prime b^\prime=abh$ , where $h=(b^{-1}hb)h_2\in H$ by the closure of

under multiplication in

. 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

. $\qedsymbol$

The set

, where

is a normal subgroup of

, is readily seen to form a group under the well-defined binary operation of left coset multiplication (the satisfaction of each group axiom follows from that of

), and is called a quotient or factor group (more specifically the quotient of

). We conclude with several examples of specific quotient groups.

Example Although the group

is not abelian, each of its subgroups its normal, so any will suffice for the formation of quotient groups; the quotient $Q_8/\langle-1\rangle$ , where $\langle-1\rangle =\{1,-1\}$ is the cyclic subgroup of

generated by

, is of order

, with elements $\langle-1\rangle ,i\langle-1\rangle =\{i,-i\},k\langle-1\rangle =\{k,-k\}$ , and $j\langle-1\rangle =\{j,-j\}$ . Since each non-identity element of $Q_8/\langle-1\rangle$ is of order

, it is isomorphic to the Klein

-group

. Because each of $\langle i\rangle$ , $\langle j\rangle$ , and $\langle k\rangle$ has order

, the quotient of

by any of these subgroups is necessarily cyclic of order

Example The center of the dihedral group

of order

(with presentation $\langle r,s\mid r^6=s^2=1,r^{-1}s=sr\rangle$ ) is $\langle r^3\rangle =\{1,r^3\}$ ; the elements of the quotient $D_6/\langle r^3\rangle$ are $\langle r^3\rangle$ , $r\langle r^3\rangle =\{r,r^4\}$ , $r^2\langle r^3\rangle =\{r^2,r^5\}$ , $s\langle r^3\rangle =\{s,sr^3\}$ , $sr\langle r^3\rangle =\{sr,sr^4\}$ , and $sr^2\langle r^3\rangle =\{sr^2,sr^5\}$ ; because

$\displaystyle sr^2\langle r^3\rangle r\langle r^3\rangle =sr^3\langle r^3\rangl... ...^3\rangle \neq sr\langle r^3\rangle =r\langle r^3\rangle sr^2\langle r^3\rangle$ ,

$D_6/\langle r^3\rangle$ is non-abelian, hence must be isomorphic to