How to prove bijection between cosets of centralizer of $g$ and element of conjugacy class of $g$. I need little help in proving this bijection
Let $G$ be any group. 
Consider the centralizer of $g \in G$ defined by
$$Zg=\{h \in G:hgh^{-1}=g\}$$  i.e. the set of all elements that commute with $g$.
Also in Group $G$, elements $g$ and $h$ are conjugates if $g = khk^{-1} : k \in G$. The set of all elements conjugate to a given $g$ is called the conjugacy class of $g$ and is denoted by $Kg$.

Prove that there is a bijection between cosets of $Zg$ and elements of $Kg$.


This follows from the orbit-stabilizer theorem. Here $G$ acts on itself by conjugation. The orbit-stabilizer theorem gives a bijection between, the cosests of the stabilizer subgroup of an element $g$, and the elements in the orbit of $g$. The stabilizer subgroup of an element $g$ is just $Zg$ and the elements in the orbit of $g$ are the elements in the conjugacy class of $g$.
