Left Coset, Order of a Group, and Index of a Subgroup

Let $(G,\ast)$ be a group, written multiplicatively as $ab=a\ast b$, and let $H$ be a subgroup of $G$.

For $a\in G$, the left coset of $H$ determined by $a$ is the subset of $G$ given by

$$aH=\{ah:\ h\in H\}.$$

The collection of all left cosets of $H$ in $G$ is denoted

$$G/H=\{aH:\ a\in G\};$$

its elements are subsets of $G$.

The sets $G$ and $G/H$ are nonempty by claim 3 of Uniqueness of the Identity Element and of Inverses in a Group, and $H$ is nonempty by condition 1 of Subgroup; the numbers of elements appearing below are therefore defined whenever the corresponding sets are finite.

If $G$ is finite, we call $G$ a finite group and call its number of elements $|G|$ the order of $G$. If $G/H$ is finite, its number of elements is called the index of $H$ in $G$ and is denoted by

$$[G:H]=|G/H|.$$