Left and Right Cosets

left/right cosets

For any $N \leq G$ (see Subgroup) and any $g \in G$ let:

g N = {g n | n \in N}; N g = {n g | n \in N}

be called (respectively) a left coset and a right coset of $N$ in $G$ . Any element of a coset is called a representative for the coset.

We denote the set of left cosets by $\frac{G}{H}$ , as we see in Quotient Group.

Warning

In general $g H \neq H g$ for all $g \in G$ . Essentially, once the left/right cosets agree with each other, then we have a Normal Subgroup specifically for that $H$ .

Examples

Say we have $G = S_{3}$ (see Symmetric Group & Permutations) and $H = ⟨ (1 2) ⟩ {1, (1 2)}$ . Then the left-cosets of $H$ are;

$g \in G$	$g H$
1	${1 1, 1 (1 2)} = {1, (1 2)} = H$
$(1 2)$	${(1 2) 1, (1 2) (1 2)} = {(1 2), 1}$
$(1 3)$	${(1 3), (1 2 3)}$
$(2 3)$	${(2 3), (1 3 2)}$
$(1 2 3)$	${(1 2 3), (1 3)}$
$(1 3 2)$	${(1 3 2), (2 3)}$
Notice that these are partitioned. Namely name:

$1 H, (1 2) H = C_{1}$
$(1 3) H, (1 2 3) H = C_{2}$
$(2 3) H, (1 3 2) H = C_{3}$
creating a partition of 3 elements.

Note

For one right coset see the order matters:

H (1 3) = {1 (1 3), (1 2) (1 3)} = {(1 3), (1 3 2)} \neq (1 3) H