What are all the left cosets of $\langle18\rangle$ in $\mathbb{Z}_{26}$.
According to Lagrange Theorem, if $G$ is a finite group and $H$ is a subgroup of $G$ then $[G:H]=\frac{|G|}{|H|}$ So $|G|=26$ and $|H|=13$ right? Meaning $[G:H]=2$. So would the 2 distinct left cosest just be,
$\{0,18,10,2,20,12,4,22,14,6,24,16,8\}$
$\{1, 19, 11, 3, 21, 13, 5, 23, 15, 7, 25, 17, 9\}$
by taking $0 + \langle18\rangle$ and $1+\langle18\rangle$.