For some reason I am always getting confused which one is the right coset, which one is the left coset.
It depends from which side you look at it.
Let’s write it down, maybe this mechanical memory will help.

So, given a group $G$, and its subgroup $H$, consider an element $g\in G$.
Then, the left coset of $H$ in $G$ is denoted by
$gH = \{gh: h\in H \}$;
and the right coset of $H$ in $G$ is denoted by
$Hg = \{gh: h\in H \}$.

As I written it down it makes sense. This is the coset of the set $H$!
It get’s translated around by an element $g$ either from the left (thus we get left cosets),
or from the right (producing right cosets).

Advertisements