Left Cosets Partition a Group and All Have the Same Cardinality

Let $(G,\ast)$ be a group, written multiplicatively, with identity element $e_G$ and inverses $a^{-1}$ as in Uniqueness of the Identity Element and of Inverses in a Group, and let $H$ be a subgroup of $G$. For $a\in G$ let $aH$ be the 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 bijection from $H$ onto $aH$. In particular, if $H$ has $t$ elements for some natural number $t$, then every left coset $aH$ has $t$ elements.