Cosets measure how a subgroup sits inside a group. Quotient groups exist only when the subgroup is normal, and this is the first place where noncommutativity changes the rules.
Left and right cosets
Definition. For \(H\le G\) and \(g\in G\), the left and right cosets are \[ gH=\{gh:h\in H\},\qquad Hg=\{hg:h\in H\}. \] Cosets are subsets of \(G\), not usually subgroups.
Proposition. The left cosets of \(H\) partition \(G\). Moreover \(gH=kH\) iff \(k^{-1}g\in H\).
Proof. Every element lies in its own coset. If \(gH\cap kH\ne\varnothing\), write \(gh_1=kh_2\). Then \(k^{-1}g=h_2h_1^{-1}\in H\), so \(g=kh\) for some \(h\in H\), and hence \(gH=kH\).
Lagrange theorem
Theorem. If \(G\) is finite and \(H\le G\), then \(|H|\) divides \(|G|\). The number of left cosets is the index \([G:H]=|G|/|H|\).
Proof. Each left coset has exactly \(|H|\) elements, since \(h\mapsto gh\) is a bijection \(H\to gH\). The cosets partition \(G\), so \(|G|=[G:H]|H|\).
Counterexample. The converse of Lagrange theorem is false. The alternating group \(A_4\) has order \(12\), but it has no subgroup of order \(6\). Such a subgroup would have index \(2\), hence be normal, forcing a quotient of order \(2\), which is incompatible with the normal subgroup structure of \(A_4\).
Normal subgroups
Definition. A subgroup \(N\le G\) is normal, written \(N\trianglelefteq G\), if \(gNg^{-1}=N\) for all \(g\in G\). Equivalently, left and right cosets agree: \(gN=Ng\) for all \(g\).
Example. Every subgroup of an abelian group is normal. The subgroup \(A_n\trianglelefteq S_n\) is normal because it is the kernel of the sign homomorphism \(\operatorname{sgn}:S_n\to\{\pm1\}\).
Counterexample. In \(S_3\), the subgroup \(H=\{e,(12)\}\) is not normal. For instance \((123)(12)(123)^{-1}=(23)\notin H\).
Constructing quotient groups
Theorem. If \(N\trianglelefteq G\), the set \(G/N\) of cosets becomes a group under \[ (gN)(hN)=(gh)N. \] The projection \(\pi:G\to G/N\), \(\pi(g)=gN\), is a surjective homomorphism with kernel \(N\).
Proof. The only delicate point is well-definedness. If \(g'=gn_1\) and \(h'=hn_2\), then \(g'h'=g h(h^{-1}n_1h)n_2\), and the middle factor lies in \(N\) by normality. Thus \(g'h'N=ghN\). The group axioms then descend from \(G\).
A quotient calculation
Calculation. In \(\mathbb Z\), the subgroup \(n\mathbb Z\) is normal and \[ \mathbb Z/n\mathbb Z=\{0+n\mathbb Z,1+n\mathbb Z,\ldots,(n-1)+n\mathbb Z\}. \] Addition of cosets is exactly addition modulo \(n\). This example is the model for quotienting by a congruence relation.