On the Fitting length of \(H_{pq}(G)\) (Q1121382)

For a natural number \(n\) the generalized Hughes subgroup \(H_ n(G)\) of a group \(G\) is defined as \(H_ n(G)=\langle g\in G\mid g^ n\neq 1\rangle\). The following result is proved: Theorem I: Let \(G\) be a finite, solvable group, \(p\) and \(q\) two primes and \(n=p\cdot q\). Then the Fitting length of \(H_ n(G)\) is at most 3, if \(H_ n(G)\neq G\). This theorem is obtained as an immediate consequence of Theorem 2: Let \(G\) be a finite, solvable group, \(H\) a proper, normal subgroup of \(G\) such that the order of every element of \(G\setminus H\) divides \(p\cdot q\), where \(p\) and \(q\) are two distinct prime numbers. Then the Fitting length of \(H\) is at most 3. This bound is best possible.