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

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

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

Left Coset of H by g is gH.

${\displaystyle \forall \;g\in G:{\color {OliveGreen}g}H=\lbrace {\color {OliveGreen}g}\ast h\;|\;h\in H\rbrace }$

Right Coset of H by g is Hg.

${\displaystyle \forall \;g\in G:H{\color {OliveGreen}g}=\lbrace {h\ast \color {OliveGreen}g}\;|\;h\in H\rbrace }$

Notice that such cosets are not necessarily subgroups of G.