Index of a Subgroup

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 $G$:

(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.