Question on the subgroups and its cosets I am studying group theory on my own with the available resources online esp. wikipedia so please be kind.
I know that a subgroup of a group is isomorphic to any of its cosets.
The map
$$f:H\rightarrow xH$$
where $H$ is a subgroup of some group and $xH$ is a coset of $H$, is an isomorphism given by sending an element $h \in H$ to $xh \in xH$.
To prove this, $f$ is one-to-one because for every $h \in H$, $xh \in xH$ is a unique element. And because the order of $H$ equals that of $xH$, this is a surjection.
I was trying to prove $f$ is an isomorphism using the following method usually used:
$f(ab)=f(a)f(b)$.
This is how I did it that I cannot complete:
$f(h_1h_2)=xh_1h_2$ ($f$ sends an element $h \in H$ to $xh \in xH$)
What should I do next so that
$f(h_1h_2)=f(h_1)f(h_2)$?

First of all that is not a isomorphism. That is just a set bijection. A coset is just a set, it has no group structure. So you cannot define a homomorphism. $f$ is just a bijection.
This bijection says that $|H|=|aH|$ for all $a \in G$.In case of finite groups, this result will later be used to prove the all- important Lagrange's therem.
