> [!NOTE] **Definition** (Left & Right Coset) >Let $(G,\circ)$ be a [[Groups|group]]. Let $(H,\circ)$ be a [[Subgroup|subgroup]] of $G.$ Let $a\in G.$ > >The left coset of $H$ containing $a$ (or the left coset of $H$ modulo $a$) is $aH=\{ a\circ h \mid h\in H\}$ > >The right coset of $H$ containing $a$ (or the right coset of $H$ modulo $a$) is $Ha=\{ h\circ a \mid h\in H \}$ **Notation:** if $H$ is [[Normal Subgroup|normal subgroup]] of $G$ then $aH=Ha.$ if $H$ is an ideal then we write $a+H$ instead on $aH.$ > [!Example] > $3+2\mathbb{Z}$ is a left coset of the subgroup $2\mathbb{Z}$ of $(\mathbb{Z},+)$ modulo $3.$ > # Properties See [[Necessary Condition for Equality of Cosets]]. # Applications [[Coset space]].