Basic doubt about cosets Studying some basic group theory I had the following doubt: For $H$ subgroup of a finite group $G$ (doesn't matter invariance of $H$), is it true that $$|G/H|=|\{aHa^{-1}:a \in G\}| \space (\text{where G/H is the quotient of G by H})?$$
I've tried to define the most natural function between these two sets that comes to my mind, $$f:G/H \to \{aHa^{-1}:a \in G\}$$$$aH \to aHa^{-1}$$
but I couldn't show that $f$ is well defined and that it is injective. I would appreciate if someone could tell me whether these two sets have the same number of elements or not.

No, it is not true. In fact, if $H$ is a normal subgroup of $G$, then (by the definition of normal) we have that for any $a\in G$, 
$$aHa^{-1}=H$$
so that
$$\{aHa^{-1}:a\in G\}=\{H\}$$
has exactly one element, whereas there is more than one element of $G/H$ as long as $G\neq H$.
