problem in meaning of fixing every element of maximal subgroup

Question

I'm not shore what is important in my question so sorry if there is irrelevant information.
Let $G$ be finite p-group (p is odd prime) and $M=C_G(u)$ is maximal subgroup of $G$ when $u \in \Omega_1(Z_2(G))-Z(G)$ and take $x \in G-M$ then define map
$\phi_u:G\Rightarrow G$ when $\phi_u(x^im)=(xu)^im$, for all $1\leq i \leq p-1$ and for all $m\in M$
I want to show that $\phi_u$ fix every element of M
but as far I see $x^im$ is not member of $M$ since $x^i \not =e $ so how can $\phi_u$ fix every member of $M$

Answer 1

I think, $i$ should be between $0\leq i \leq p-1$, otherwise your function $\phi_u$ is not from $G$ to $G$ (as $x^im\notin M$ when $1\leq i \leq p-1$ ).

On the other hand, $\phi_u(m)=\phi_u(x^0m)=(xu)^0m=m$ for all $m\in M$.