Finite soluble groups with metabelian centralizers. (Q471886)

Let \(G\) be a finite solvable group. The authors analyze what can be said about \(G\) when we know that the centralizer in \(G\) of every non-trivial element of \(G\) satisfies some given property. More specifically, they define \(G\) to be in \(CA_d\) if the derived length of the centralizer of every non-identity element of \(G\) is at most \(d\). In general, they prove that any group in \(CA_d\) has derived length at most \(d(3d+2)\) and Fitting height at most \(3d+2\). When \(d=2\) then the derived length of \(G\) is at most \(4\). When the centralizer of every non-identity element of \(G\) is such that its derived subgroup is nilpotent then \(G^{(3)}\) is nilpotent.