TITLE: Size of |G:H| in set theory
QUESTION [0 upvotes]: Let $|G:H|$ be the set of left cosets for $x\in G$ and $xh\in xH$.
It is stated that for all $g\in G$, $g$ is in exactly $1$ coset, as for $g,e \in G, ge=g\in gH$.
But I don't then see how it isn't the case that $|G:H|=|H|$, despite $|G|=|G:H||H|$.
I'm confused on this. Please help, in general.

REPLY [1 votes]: Take this;
$G=\mathbb Z$ and $H= 2\mathbb Z$ then $[G:H]=2$
;
 is it equal to $H$?
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