Coset

For a Subgroup $H\le G$ and an element $g\in G$:

• The left coset of $H$ by $g$ is $gH=\{gh:h\in H\}\subseteq G$.
• The right coset is $Hg=\{hg:h\in H\}\subseteq G$.

In an abelian group (or for normal subgroups in general acted on a particular way) left and right cosets coincide. In general they need not.

Properties

1. Either $gH=g^{\prime }H$ or $gH\cap g^{\prime }H=\varnothing$ - cosets partition $G$.
2. $g_{1}H=g_{2}H⟺g_1^{-1}{g}_2\in H$.
3. All cosets have the same size: $|gH|=|H|$.
4. The number of distinct cosets is the index $[G:H]=|G|/|H|$ (for finite $G$).

(Fact 3 + the partition give Lagrange’s theorem: $|H|\mid |G|$.)

Quotient Construction

If $H$ is Normal subgroup ($H⊴G$), the cosets form a group under $(g_{1}H)(g_{2}H)=(g_1{g}_2)H$ - the Quotient group $G/H$.

For non-normal $H$ this product is not well-defined: two different representatives $g_{1}h$ and $g_1{h}^{\prime }$ give different products $(g_{1}h\cdot g_2)H$.

Geometric Picture

Cosets of $H$ in $G$ are the “horizontal slices” of $G$ partitioned by the equivalence $g_1\sim g_2⟺g_1^{-1}{g}_2\in H$. Each slice has the same size $|H|$, and there are $[G:H]$ of them.

Examples

• $G=\mathbb{Z},H=2\mathbb{Z}$: two cosets, the even integers $0+2\mathbb{Z}$ and the odd integers $1+2\mathbb{Z}$.
• $G=S_3,H=⟨(12)⟩$: three left cosets $\{e,(12)\},\{(123),(13)\},\{(132),(23)\}$. Right cosets are different.
• $G=\mathbb{R},H=\mathbb{Z}$: cosets are $r+\mathbb{Z}$ for $r\in [0,1)$ - the quotient $\mathbb{R}/\mathbb{Z}\cong S^1$ (the circle).