Index of a Subgroup - Knowlpedia

Let $H$ be a subgroup of a group $G$ . The index of $H$ in $G$ , denoted $[G:H]$ , is the cardinality of the set of left cosets of $H$ in $G$ :

[G:H] := \bigl|\{gH : g\in G\}\bigr|.

(Equivalently, it is the number of distinct right cosets.)

For finite $G$ , Lagrange's theorem implies $[G:H] = |G|/|H|$ , so in particular $|H|$ divides $|G|$ . Small index has strong consequences: $[G:H]=1$ iff $H=G$ , and if $[G:H]=2$ then $H$ is normal, i.e. a normal subgroup.

Examples

In $\mathbb{Z}$ with $H=3\mathbb{Z}$ , one has $[\mathbb{Z}:3\mathbb{Z}]=3$ since there are exactly three residue classes mod $3$ .
In $S_3$ with $H=\{e,(12)\}$ , one has $[S_3:H]=3$ because $|S_3|=6$ and $|H|=2$ .
In an infinite group, the index can be infinite; for example, $[\mathbb{Z}:2\mathbb{Z}] = 2$ but $[\mathbb{Z}:\{0\}]$ is infinite.