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

Given a group $G$ and a subgroup $H$ of $G$ , the relation $\sim_L$ 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 $ab^{-1}\in H$ is called right congruence modulo $H$ (observe that these two relations coincide if $G$ is abelian).

Proof. We will only give the proof for left congruence modulo $H$ , as the argument 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 reflexive. If $b\in G$ satisfies $a\sim_L b$ , so that $b^{-1}a\in H$ , then by the closure 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 transitive, hence an equivalence relation.

It follows from the preceding proposition 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 $\bar{a}$ the equivalence class containing $a$ under $\sim_L$ , we see that \begin{equation*} \begin{split} \bar{a}&=\set{b\in G\mid b\sim_L a}\\ &=\set{b\in G\mid a^{-1}b\in H}\\ &=\set{b\in G\mid b=ah\text{ for some }h\in H}=\set{ah\mid h\in H}\text{.} \end{split} \end{equation*} 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 $a+H=\set{a+h\mid h\in H}$ and $H+a=\set{h+a\mid h\in H}$ ). 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 $\abs{G:H}$ , 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 $\abs{G:H}$ 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 product 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 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 $H$ 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 $a=1$ , $a^\prime=h$ , 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 $g$ gives $ghg^{-1}=h^\prime\in H$ , and because $g$ and $h$ 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 $H$ 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 $a^\prime=ah_1$ and $b^\prime=bh_1$ ; now, we have \begin{equation*} a^\prime b^\prime=ah_1bh_2=a(bb^{-1})h_1bh_2 =ab(b^{-1}h_1b)h_2\text{,} \end{equation*}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 $H$ under multiplication 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 satisfaction of each group axiom 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.