A question about Theorem 2.2 in Gorenstein's book and a finite $p$-group

Question

Any comment about the following two questions will be greatly appreciated!

According to the Theorem 2.2 in page 224 of Gorenstein's book, finite groups, we have

Let $A$ be a $\pi^{\prime}$-group of automorphisms of the $\pi$-group $G$, and suppose $G$ or $A$ is solvable. Then for each prime $p$ in $\pi$, $A$ leaves invariant some Sylow $p$-subgroup $P$ of $G$. My first question is

1. Could the action of $A$ on $P$ be trivial?(I mean is it possible $P\rtimes A=P\times A$?)

I have another question about a finite $p$-group $P$.

2. Let $P$ be a $p$-group, whose all maximal subgroups are cyclic. How could we show that the nilpotency class of $P$ is at most two?

Many thaks!

Answer 1

The answer to the first question is yes, which is quite possible.

Firstly we can assume that it has more than one maximal subgroup. Let $H,K$ be cyclic maximal subgroups of $G$. Note that $H\cap K\leq Z(G)$ as $H\cap K$ both commutes with elements of $H$ and $K$ which leads that $H\cap K$ commutes with elements of $HK=G$.

Notice that $G/H\cap K$ is abelain as $\overline G=\overline H\times \overline K$.