Transfer normal Fitting pairs do not always suffice

AbstractBerger (Proc. London Math. Soc. (3) 42 (1981), 59–86) gave a method for determining X∗, where X is a Fitting class obeying certain conditions. In particular he showed that X∗ could be determined by certain transfer normal Fitting pairs. This provided for the first time a method for actually determining whether or not a particular group G was a member of the class X∗, a class whose existence was predicted in a theoretical manner not amenable to actual computation. In this paper we present an example of a class X where X∗ cannot be determined using Berger's transfer normal Fitting pairs. This example provides important limitations on the power of Berger's methods, and suggests how Berger's theorem should be strengthened. In order to build this example it was necessary to construct new techniques, similar to those of Dark (Math. Z. 127 (1972), 145–156), but differing in that more than one generating group can be used, and these may have arbitrary p-groups acting at the top instead of just cyclic group of order p. Since techniques for the construction of examples are extremely important in Fitting class theory, the general techniques developed here are probably of greater value than the specific example for which they were constructed