TL;DR at bottom
The problem statement is as follows:
Prove the following corollary without the assumption that $G$ is finite. For $K < H < G$ $$[G:K] = [G:H][H:K]$$ Specifically, suppose $\mathcal{H} \subset G$ and $\mathcal{K} \subset H$ are coset representatives for $H < G$ and $K < H$ respectively and prove that $$\mathcal{J} = \{gh \mid g \in \mathcal{H},\ h \in \mathcal{K} \}$$ is a set of coset representatives for $K$ in $G$. Consequently, $$[G:K] = \lvert \mathcal{J} \rvert = \lvert \mathcal{H} \rvert \lvert \mathcal{K} \rvert = [G:H][H:K]$$
I know the problem says that $\mathcal{H}, \mathcal{K}$ are coset representatives, but I'm assuming they meant to say that sets of coset representatives. Given that we can interpret these sets to be $$\mathcal{H} = \{ g \in G \mid \text{for any}\ g'H \subset G,\ g'H = gH \}$$ $$\mathcal{K} = \{h \in H \mid \text{for any}\ h'K \subset H,\ h'K = hK \}$$ In other words, $\mathcal{H,K}$ are sets of elements of $G,H$ such that the coset constructed from the elements $g \in G, h \in H$ are equivalent to any other left coset in the set of cosets $G/H$ and $H/K$. Correct? I've adapted these definitions from the following one given by my text:
If $H<G$, a collection of elements $\mathcal{H} \subset G$ is called a set of left coset representatives for $H$ in $G$ if for each left coset $g'H \subset G$ there exists exactly one $g \in \mathcal{H}$ such that $g'H = gH$
I'm having some trouble gaining intuition for this concept since it seems this definition is in conflict with an example in the section that states that the unique integers $r \in \{0, 1, 2, ..., n-1 \}$ are the coset representatives of the elements $[a] \in \mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$ since for any $[a]$ we have $[r] = [a]$. Initially I thought that this definition was saying that for all cosets in $G/H$ there was only one coset, call it $gH$ that was equivalent to them all. This clearly can't be correct though since $[1] \neq [4]$ for $\mathbb{Z}_6$, say. So, is it just the case that for a subset of cosets of a certain set of cosets that are equivalent then there is one coset that is equivalent to them all? I.e., for say the congruence class of $7$ in $\mathbb{Z}_6$ the coset representative would be $[1]$ since $$[1] = \{1 + 6k \mid k \in \mathbb{Z} \} = \{..., 1, 7, 13, 19,... \}$$ But couldn't $[7]$ fulfill that role just as well? Since in the definition of the congruence class indicates $k \in \mathbb{Z}$ and as such can take on negative values so for $$[7] = \{7 + 6k \mid k \in \mathbb{Z} \} = \{..., 1, 7, 13, 19,... \}$$ So yes we see that $[1] = [7]$ but what makes $[1]$ special and fit to fulfil the definition of a coset representative but not $[7]$ or $[13]$, etc..?
TL;DR
Having trouble gaining an intuition for the concept of (a set of) coset representatives and as such, unable to really put the pieces together for this problem.
