Let $(G,\ast)$ be a \reftext{def:group-2026a}{group}, written multiplicatively as $ab=a\ast b$, and let $H$ be a \reftext{def:subgroup-2026a}{subgroup} of $G$.

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

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

its elements are subsets of $G$.

The sets $G$ and $G/H$ are nonempty by claim 3 of \ref{thm:group-identity-inverse-uniqueness-2026a}, and $H$ is nonempty by condition 1 of \ref{def:subgroup-2026a}; the numbers of elements appearing below are therefore defined whenever the corresponding sets are \reftext{def:finite-set-2026a}{finite}.

If $G$ is finite, we call $G$ a \textbf{finite group} and call its \reftext{def:number-of-elements-2026a}{number of elements} $|G|$ the \textbf{order} of $G$. If $G/H$ is finite, its number of elements is called the \textbf{index} of $H$ in $G$ and is denoted by