A Fitting height lemma and its applications

Abstract

Let A be a group acting on a solvable group G and let N be an A-invariant normal subgroup of G such that $$[G,A]\nsubseteq N$$ . We prove the inequality $$h([G,A])\le h([G,A]N/N)+h([N,A])$$ where h(G) denotes the Fitting height of G. As an application of this result, we obtain several Fitting height inequalities. A new concept “fixed point free separability” and a new characteristic subgroup Y(G) is defined and used in order to prove some further results about the Fitting height of a group. In the last section, a new characterization of solvable groups is given: a group G is solvable if and only if it is fixed point free separable.