Let $(G,\ast)$ be a \reftext{def:group-2026a}{group}, written multiplicatively, with identity element $e_G$ and inverses $a^{-1}$ as in \ref{thm:group-identity-inverse-uniqueness-2026a}, and let $H$ be a \reftext{def:subgroup-2026a}{subgroup} of $G$. For $a\in G$ let $aH$ be the \reftext{def:left-coset-order-index-2026a}{left coset} determined by $a$. Then for all $a,b\in G$ the following hold.

1. $a\in aH$. In particular every element of $G$ lies in at least one left coset, and $H=e_GH$ is itself a left coset.
2. $aH=bH$ if and only if $a^{-1}b\in H$.
3. If $aH\cap bH\ne\emptyset$, then $aH=bH$. Consequently every element of $G$ lies in exactly one left coset of $H$.
4. The map $h\mapsto ah$ is a \reftext{def:bijection-sets-2026a}{bijection} from $H$ onto $aH$. In particular, if $H$ \reftext{def:number-of-elements-2026a}{has $t$ elements} for some \reftext{def:natural-numbers-2026a}{natural number} $t$, then every left coset $aH$ has $t$ elements.