A Fitting height lemma and its applications
Let \(A\) be a group acting on a finite solvable group \(G\) and let \(N\) be an \(A\)-invariant normal subgroup of \(G\) such that \([G,A]\not\subseteq N.\) The author proves the inequality \(h([G,A])\leq h([G,A]N/N) + h([N,A])\) where \(h(G)\) denotes the Fitting height of \(G.\) As an application of this result, he obtains several Fitting height inequalities, involving the new concept of ``fixed point free separability''. An \(A\)-invariant subnormal series of \(G,\) \(1=N_k \unlhd \cdots \unlhd N_0=G,\) is called a fixed point free series of \(G\) with respect to \(A\) if the action of \(A\) on each quotient \(N_i/N_{i+1}\) is either trivial or fixed point free. The group \(G\) is said to be fixed point free separable if there is a fixed point free series of \(G\) with respect to each subgroup \(A\) of Aut\((G)\). A new characterization of solvable groups is given: a finite group \(G\) is solvable if and only if it is fixed point free separable.
scientific article; zbMATH DE number 7371066
scientific article