the generalized Fitting subgroup contains its own centralizer. This means that in some sense the generalized Fitting subgroup controls G, because G modulo the centralizer of F*(G) is contained in the automorphism group of F*(G), and the centralizer of F*(G) is contained in F*(G). In particular there are only a finite number of groups with given generalized Fitting subgroup. Applications The normalizers of nontrivial p-subgroups of a finite group are called the p-local subgroups and exert a great deal of control over the structure of the group (allowing what is called local analysis). A finite group is said to be of characteristic p type if F*(G) is a p-group for every p-local subgroup, because any group of Lie type defined over a field of characteristic p has this property. In the classification of finite simple groups, this allows one to guess over which field a simple group should be defined. Note that a few groups are of characteristic p type for more than one p. If a simple group is not of Lie type over a field of given characteristic p, then the p-local subgroups usually have components in the generalized Fitting subgroup, though there are many exceptions for groups that have small rank, are defined over small fields, or are sporadic. This is used to classify the finite simple groups, because if a p-local subgroup has a known component, it is often possible to identify the whole group . The analysis of finite simple groups by means of the structure and embedding of the generalized Fitting subgroups of their maximal subgroups was originated by Helmut Bender and has come to be known as Bender's method. It is especially effective in the exceptional cases where components or signalizer functors are not applicable. References Finite groups Functional subgroups