Originally Posted by

**Amer** find all left coset taking the subgroup generated by $\displaystyle <(1,2)>$

the group is $\displaystyle \mathbb{Z}_2\times \mathbb{Z}_4 $

we have the subgroup $\displaystyle h={(0,0),(1,2)} $

$\displaystyle \mathbb{Z}_2\times \mathbb{Z}_4 =[(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)]$

I take all elements not in h and add them to h to get the left coset but I have repeated elements so how I can chose elements such that I do not have repeated ones

the book said

$\displaystyle h+(0,0) = h $

$\displaystyle h+(1,0) $

$\displaystyle h+(0,1)$

$\displaystyle h+(1,1) $

how the book take these elements can you help

is there a way to chose these elements (Thinking)