Struggling to understand where I'm going wrong here. I have $ G=S_3 $ and $ H = \{e, (1 2)\} $, and I want to list all the cosets with respect to H. This is what I have so far:
Left cosets
$eH=\{e, (12)\}=H$
$(12)H=\{(12),(12)(12)=e\}=\{(12), e\}$
$(13)H=\{(13),(13)(12)=(123)\}=\{(13),(123)\}$
$(23)H=\{(23),(23)(12)=(132)\}=\{(23),(132)\}$
$(123)H=\{(123),(123)(12)=(13)\}=\{(123),(13)\}$
$(132)H=\{(132),(132)(12)=(23)\}=\{(132),(23)\}$
Right cosets
$He=H$
$H(12)=\{(12),(12)(12)=e\}=\{(12), e\}$
$H(13)=\{(13),(12)(13)=(132)\}=\{(13),(132)\}$
$H(23)=\{(23),(12)(23)=(123)\}=\{(23),(123)\}$
$H(123)=\{(123),(12)(123)=(23)\}=\{(123),(23)\}$
$H(132)=\{(132),(12)(132)=(13)\}=\{(132),(13)\}$
I've reviewed all my calculations, and they seem right, but this results in 5 distinct sets when there should be $ \frac{|G|}{|H|} = \frac{3!}{2} = 3$. What am I doing wrong?