On the Fitting length of NC-groups (Q1176347)

A finite group \(G\) is called an NC-group if, for each prime \(p\), the normalizer and the centralizer of the centre of the Sylow \(p\)-subgroups of \(G\) coincide. Soluble NC-groups with arbitrarily large Fitting length and trivial centre are constructed. Theorem 4 is central to the paper: If \(H\) is a non-identity \(\{p,q\}\)-group such that \(O_ p(H)=1\) and if a Sylow \(p\)-subgroup of \(H\) is self-normalizing, then there is a \(\{p,q\}\)- group \(G\) satisfying (1) \(Op(G)=1\); (2) Sylow \(p\)-subgroups of \(G\) are self-normalizing; (3) \(G/F_ 2(G)\cong H\), where \(F_ 2(G)\) denotes the second term of the Fitting series of \(G\), as usual; (4) \(G\) is an NC- group; (5) the Fitting length of \(G\) exceeds that of \(H\) by 2, while \(p\)- length and \(q\)-length of \(G\) each exceed that of \(H\) by 1; (6) the centre of \(G\) is trivial; (7) \(G\) is primitive. Using this theorem and induction on \(n\), the authors prove Theorem 5: Let \(n\) be an integer greater than 2. Then there exists a \(\{p,q\}\)-group \(G_ n\) of Fitting length \(2n\), both its \(p\)-length and its \(q\)-length being \(n\), with trivial centre. Moreover, \(O_ p(G_ n)=1\), the Sylow \(p\)-subgroups of \(G_ n\) are self-normalizing, and \(G_ n\) is an NC-group. If the generalized Fitting subgroup of a finite group \(G\) is simple nonabelian, then \(G\) is not an NC-group (Theorem 2.2).