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

< Abstract Algebra | Group Theory | Subgroup

< Abstract Algebra | Group Theory | Subgroup

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

Left Coset of H by *g* is *g*H.

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

Right Coset of H by *g* is H*g*.

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

Notice that such cosets are not necessarily subgroups of G.