Quotient group

The elements of \(G/N\) are the cosets \(aN\), each treated as a single object. The product \((aN)(bN)=(ab)N\) is well defined because \(N\) is normal: if \(aN=a'N\) and \(bN=b'N\), then \(a'b'N=abN\), so the result does not depend on the representative. The identity is \(N\) and the inverse of \(aN\) is \(a^{-1}N\).

The most familiar case is \(\mathbb{Z}/n\mathbb{Z}\): the cosets of \(n\mathbb{Z}\) are the residues mod \(n\), and adding cosets is modular addition. So \(\mathbb{Z}_{12}/\{0,4,8\}\) has four elements and its cosets add like residues mod 4. In general \(|G/N|=[G:N]=|G|/|N|\) when \(G\) is finite.

The extreme cases serve as a check: if \(N=\{e\}\) the quotient is a copy of \(G\), and if \(N=G\) a single coset remains, the trivial group. If \(G\) is abelian, so is \(G/N\); in \(S_3/A_3\) two elements remain, one for each «parity» of permutation.

Watch out for two frequent mistakes: thinking \(G/N\) is a subgroup of \(G\) (its elements are sets, not elements of \(G\)) and writing \(G/N\) when \(N\) is not normal, where the product of cosets is not well defined. Later the first isomorphism theorem will show that every quotient appears as the image of a homomorphism.