Hw8 Problem 2

(a) Find all the left cosets of the subgroup $\{\rho_0, \mu_2\}$ of $D_4$. (Table 8.12 gives the multiplication table for $D_4$.)

(b) Now find the right cosets. Are they the same as the left cosets?

Solution

(a)
$\{\rho_0, \mu_2\}$
$\{\rho_1. \delta_2\}$
$\{\rho_2, \mu_1\}$
$\{\rho_3, \delta_1\}$

(b)
$\{\rho_0, \mu_2\}$
$\{\rho_1, \delta_1\}$
$\{\rho_2, \mu_1\}$
$\{\rho_3, \delta_2\}$

No, they are not the same.