Elements in the same coset and the Cayley Diagram

Question

A question from Visual group theory, by Nathan Carter. In a Cayley diagram, if $aH$ is a coset of a subgroup $H$ of a group $G$ and $b$ belongs to $aH$, why is it that every node that can be reached from $a$ using generators of subgroup $H$ can also be reached from $b$ using those same generators?

Answer 1

The points which can be reached from $a$ by a product of generators of $H$ are precisely the point in $\{a h\mid h\in H\}$ because every element in $H$ can be written as a product of generators of $H$ (by definition of generators). This set is equal to $aH$ and so as $b\in aH$ it follows that $b$ can be reached by generators from $H$. This is essentially saying that $aH=bH$ for all $b\in aH$.

Let's make this slightly more explicit. Suppose $b=ah$ for some $h\in H$. Consider the point $c$ which we can reach from $a$ by multiplying by some element in $H$, so $c=ah'$ for some $h'\in H$. Well then $c=(bh^{-1})h'=b(h^{-1}h)$ and so we can also reach $c$ from $b$ because $h^{-1}h$ is in $H$.