Question

Find all left cosets of the subgroup {ρ₀,ρ₂} = {e,(1,3)(2,4)} in D₄

Answer 1

Remind that $D_4=\{\rho_0=e,\rho_1=(1\ 2\ 3\ 4), \rho_2=\rho_1^2, \rho_3=\rho_1^3, \sigma=(1\ 3), \sigma\rho_1, \sigma\rho_1^2, \sigma\rho_1^3\}$ , $\sigma^2=\rho_1^4=(\sigma\rho_1)^2=e$. Given an element $g\in D_4$, the left coset in $D_4$, generated by the element g and the subgroup $H=\{e,\rho_1^2\}$, is the set $\{gh:h\in H\}$ . Two left cosets gH and g'H are the same, if and only if $g'\in gH$ , if and only if $g^{-1}g'\in H$ . Let calculate.

$eH=H=\{e,\rho_1^2\}=\rho_1^2H$

$\rho_1 H=\{\rho_1,\rho_1^3\}=\rho_1^3H$

$\sigma H=\{\sigma, \sigma\rho_1^2\}=\sigma\rho_1^2H$

$\sigma\rho_1 H=\{\sigma\rho_1, \sigma\rho_1^3\}=\sigma\rho_1^3H$

The number of different left cosets is 4. This is an index of the subgroup H in $D_4$ .