Understanding Proof of Sylow theorem from Herstein I was trying to learn group theory indepedently form Herstein' Topics of Algebra.
I that I now reading Sylow's theorem.In that I trying to understand first proof of sylow theorem. But I am not able to getting motivation from that.I tried.But not able to get .I attached pic from book



Actually I not able to understand 
1)Why such equivalence class is defined ?
2)What is divisor argument mentioned in this proof?
As due to this 2 question with me I could not Proceed further.
Any Help will be appreciated

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*Multiplication with an element of a group is a permutation. Hence the defined relation is reflexive (take g=id the identity), symmetric (if $M_1$=$M_2g$, then $M_2=M_1g^{-1}$) and transitiv (if $M_1=M_2g$ and $M_2=M_3h$ then $M_1=M_3(hg)$). So the relation defines an equivalence relation and you might talk about their equivalence classes.

*The equivalence classes of a set form a partition of that set. Hence $M$ is a disjoint union of a set of representative equivalence classes and the cardinality of $M$ is the sum of the cardinalities of these representative equivalence classes. Therefore if the cardinality of each equivalence class of $M$ is divisible by $p^{r+1}$ elements, then the cardinailty of $M$ is divisible by $p^{r+1}$. But Herstein has argued before that $p^{r+1}$ cannot divide $|M|$.Hence there must  exist one equivalence class of $M$ whose cardinality is not divisible by $p^{r+1}$.
