Good action on a finite group.

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) ⩽ ℓ (A) where h (G) is the Fitting height of G and ℓ (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. [ABSTRACT FROM AUTHOR]