Q:" G group, H subgroup. G=D6, D diedras group, and H=. Find the cosets of H."
I don't understand the method to find the cosets, I've searched for answers but somehow it stays confusing…
Being aH the left coset, do I give an arbitrary value from $D_6$ to $a$? And how does it multiply by $H=\{e, r, r^2, r^3, r^4, r^5\}$?
Best Answer
There's no "method" to compute cosets, just the definition.
Given $a\in G$, the coset $aH$ is the subset of $G$ consisting of the elements of the form $ah$ as $h$ varies through $H$.
In the case under question $G=D_6$ and $H$ is the subgroup of rotations, so
if $a\in H$ the coset $aH$ is just $H$;
if $a\notin H$ the coset $aH$ consists of the 6 elements in $G$ not in $H$.
(These two claims should be checked)
A general property of cosets is that they define a partition of $G$. This can be easily verified in the above example.