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 $j_0=j_1$?


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .