Index of a subgroup

The index is written \([G:H]\) and may be finite or infinite. In \(\mathbb{Z}\), the subgroup \(n\mathbb{Z}\) has index \(n\) because its cosets are the \(n\) residues modulo \(n\); the trivial subgroup of \(\mathbb{Z}\), on the other hand, has infinite index. Every group is its own only subgroup of index \(1\).

It makes no difference whether one counts left or right cosets: the assignment \(aH\mapsto Ha^{-1}\) is a bijection between the two, so both counts agree even though the cosets, as sets, need not.

When \(G\) is finite, every coset has \(|H|\) elements and the cosets partition \(G\); hence \(|G|=[G:H]\,|H|\), and the index is the quotient \(|G|/|H|\). This equality is the content of Lagrange's theorem, which comes next.

A common mistake is to confuse the index with the order: in \(\mathbb{Z}_{12}\), the subgroup \(\{0,3,6,9\}\) has order \(4\) and index \(3\). Subgroups of index \(2\) deserve a mention: their two cosets are \(H\) and \(G\setminus H\), and later we will see they are always normal.