Abstract Algebra/Group Theory/Subgroup/Coset/Definition of a Coset

h*g is in right coset Hg if h is in subgroup H of G and g is in group G.

Let g be a fixed element of Group G. Let H be a subgroup of G.

Left Coset of H by g is gH.

$\mathrm{\forall }g\in G:gH=\{g\ast h|h\in H\}$

Right Coset of H by g is Hg.

$\mathrm{\forall }g\in G:Hg=\{h\ast g|h\in H\}$

Notice that such cosets are not necessarily subgroups of G.