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.

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.