Question about distinctness of coset given $H \leq K \leq G$

Question

This question is regarding a detail I am unsure about in a problem in the Dummit and Foote abstract algebra textbook that I am currently working on. The problem itself is that if $H \leq K \leq G$ I need to show $|G:H| = |G:K|\cdot|K:H|$. Within this problem I am not sure about the uniqueness of how the cosets are counted in the case of both sides being finite.

If $H \leq K \leq G$ as defined and I let $x_i$ for $i = 1, \dots, |G:K|$ be the coset representatives of $K$ in $G$ and $y_j$ for $j=1,\dots,|K:H|$ be the coset representatives of $H$ in $K$, why is it that each pair of cosets, $x_iy_j$ is a unique coset representative of $H$ in $G$?

Suppose $x_{i_0}y_{j_0}$ and $x_{i_1}y_{j_1}$ are two such cosets representatives where $x_{i_0}y_{j_0} = x_{i_1}y_{j_1}$. What makes it necesarry for $i_0=i_1$ and $j_0=j_1$?

Answer 1

Hint: argue $~~x_{i_0}y_{i_0}H=x_{i_1}y_{i_1}H~~~\Rightarrow~~~ x_{i_0}K=x_{i_1}K~~~\Rightarrow~~~ x_{i_0}=x_{i_1}~~~\Rightarrow~~\cdots$

In the language of set partitions, this should be "obvious." Consider a map with "nations" and within each nation are "territories." One can speak of all the territories in a single nation or all the territories in the world. Let $A$ and $B$ be two territories. If $A=B$ in the context of the whole map, then they both inhabit the same nation, and within that nation they are both the same territory.

This metaphor is applicable to $H\le K\le G$, where $G$ is the whole map, $G/K$ is the set of nations and $G/H$ is the entire map's set of territories. The systems of representatives basically amount to labels associated with each nation or territory. Or, in direct algebraic language:

$$G=\bigsqcup_i x_iK=\bigsqcup_i x_i\left(\bigsqcup_j y_jH\right)=\bigsqcup_i\bigsqcup_jx_i(y_jH)=\bigsqcup_{i,j}x_iy_jH.$$

Check that bijections (e.g. left multiplications) preserve disjoint unions so the above is valid.