Good action on a finite group

TL;DR

This paper introduces the concept of 'good action' of a finite group on another and proves a new theorem confirming a long-standing conjecture about the Fitting height of solvable groups under certain conditions, especially when both groups have odd order.

Contribution

It defines 'good action' and establishes a new noncoprime Hall-Higman type theorem, confirming the conjecture for odd order groups with good actions.

Findings

Confirmed the conjecture for odd order groups with good actions.

Proved a new noncoprime Hall-Higman type theorem.

Established conditions under which the Fitting height bound holds.

Abstract

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.