I want to translate the following theorem from German to English:
5.12 Satz. Sei $\mathfrak{P}$ eine $p$-Gruppe und $\alpha$ ein Automorphismums von $\mathfrak{P}$ von su $p$ teilerfremdder Ordnung.*
- Für $p\gt 2$ lasse $\alpha$ alle Elemente der Ordnung $p$ von $\mathfrak{P}$ fest.
- Für $p\gt 2$ lasse $\alpha$ alle Elemente der Ordnungen $2$ und $4$ von $\mathfrak{P}$ fest.
Dann ist $\alpha =1$.
I used google translator and get the following:
Let B be a p-group and S be an automorphism of B of order coprime to p.
a) For p> 2, let S fix all the elements of B of order p.
b) for p = 2, let S fix all elements of orders 2 and 4 of B.
Then S = 1
I found the following translation on some research
For a finite $p$ group $P$, we write $\Omega(P) = \left\{\begin{array}{ll} \Omega_1(P) &\text{if }p\gt 2\\ \Omega_2(P) &\text{if }p=2 \end{array}\right.$ where $\Omega_i(P) = \Bigl\langle x\in P \;\Bigm|\; |x| \bigm| p^i\Bigr\rangle.$
Lemma 2.1 ([6] Kap. IV, Satz 5.12]). Suppose that a finite $p'$-group $A$ acts by automorphisms on a finite $p$-group $P$. If $A$ acts trivially on $\Omega_1(P)$ for $p\neq 2$ or on $\Omega_2(P)$ for $p=2$,then $A$ acts trivially on $P$.
I want to know if the last translation is true because I am not sure that if an automorphism comes from a $p^{'}$-group, then this automorphism is of order coprime to $p$.