How can a family of cosets be a partition of a group?

Question

$G$ a group with subgroup $ H \leq G $.

Theorem: The family of all the cosets $Ha$ as $a$ ranges over $G$, is a partition of $G$.

I have a little bit of trouble understanding this. For something to create a partition it has to be disjoint, but how can a family of cosets be disjoint?

If we consider two cosets, lets say, $Ha$ and $Hb$ They must have an element in common, namely the identity element $e$. By definition of a subgroup; $H$ contains the identity element, but that means that $e \in Ha$ and $e \in Hb$ and $Ha$ and $Hb$ are not disjoint.

Answer 1

If we consider two cosets, lets say, $Ha$ and $Hb$ They must have an element in common, namely the identity element $e$.

This is not true : if $e \in Ha$, then there is $h\in H$ with $ha = e$, so $a = h^{-1} \in H$ and $Ha = H$.

Actually, two cosets $Ha$ and $Hb$ are either disjoint or equal : if there is $g\in Ha \cap Hb$, there are $h,h'\in H$ such that : $$g= ha = h'b$$ Then $Ha = H(h^{-1}hb)= Hb$.

Therefore, the set $\{Ha : a\in G\}$ is a partition of $G$

(Note that depending on your definition, the family $(H_a)_{a\in G}$ may not bt a partition, since each coset will appear several times)