Let $G$ and $A$ be finite groups with $A$ acting on $G$ by automorphisms. In this paper we introduce the concept of "good action"; namely we say the action of $A$ on $G$ is good, if $H=[H,B]C_H(B)$ for every subgroup $B$ of $A$ and every $B$-invariant subgroup $H$ of $G.$ This definition allows us to prove a new noncoprime Hall-Higman type theorem. If $A$ is a nilpotent group acting on the finite solvable group $G$ with $C_G(A)=1$, a long standing conjecture states that $h(G)\leq \ell(A)$ where $h(G)$ is the Fitting height of $G$ and $\ell(A)$ is the number of primes dividing the order of $A$ counted with multiplicities. As an application of our result we prove the main theorem of this paper which states that the above conjecture is true if $A$ and $G$ have odd order, the action of $A$ on $G$ is good and some other fairly general conditions are satisfied.

中文翻译：

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有限群上的好动作

令 $G$ 和 $A$ 是有限群，其中 $A$ 通过自同构作用于 $G$。在本文中，我们引入了“良好行动”的概念；即我们说 $A$ 对 $G$ 的作用是好的，如果 $H=[H,B]C_H(B)$ 对于 $A$ 的每个子群 $B$ 和每个 $B$-不变子群 $H $ of $G.$ 这个定义允许我们证明一个新的非互质霍尔-希格曼类型定理。如果 $A$ 是作用于有限可解群 $G$ 且 $C_G(A)=1$ 的幂零群，一个长期存在的猜想表明 $h(G)\leq\ell(A)$ 其中 $h( G)$ 是 $G$ 的拟合高度，$\ell(A)$ 是除以重数计算的 $A$ 阶数的素数数。作为我们结果的应用，我们证明了本文的主要定理，即如果 $A$ 和 $G$ 具有奇数阶，则上述猜想成立，