Dtyjlui

This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help..
I need to prove these following statements are equivalent :
Suppose that $P:Ylongrightarrow X $ be a covering space. Prove that these three statements are equivalent :

A) $ {P_*}(pi_1(Y,{y_0}))lhd pi_1(X,P(y_0))$

B) For any two $y_0,y_1in P^{-1}(x_0)$ we have $ {P_*}(pi_1(Y,{y_0}))= {P_*}(pi_1(Y,{y_1}))$

C) For any loop $omega$ in $X$, all of its lifts $widetildeomega$ in $Y$ are loops or none of them is a loop.

I know I should try to show the equivalency by conjugacy classes but still I can not prove it, really thank anyone who help me with solving this problem

Let me give you indications, rather than a full solution :

A) => B) : note that if $alpha $ is a path from $y_0$ to $y_1$, then $p_*(pi_1(Y,y_0))$ and $p_*(pi_1(Y,y_1))$ are conjugate with $p_*(alpha)$ (which is a loop based at $x_0$).

B)=> C) : if $omega$ lifts to a loop $tilde{omega}$, then it is in $p_*(pi_1(Y,y_0))$ for some $y_0$; and so for all $y_0$. What does it mean ?

C) => A) : let $alpha$ be a loop based at $x_0$, $beta$ a loop based at $y_0$, and consider $alpha p_*(beta)alpha^{-1}$. If you lift $alpha$ to a path $gamma$ starting at $y_0$, then you can lift $p_*(beta)$ to a loop $delta$ based at $gamma (1)$. Then what is $p_*(gammadeltagamma^{-1})$ ?