It may also be explicitly constructed as the product of the p-cores of G over all of the primes p dividing the order of G. If G is a finite non-trivial solvable group then the Fitting subgroup is always non-trivial, i.e. if G≠1 is finite solvable, then F(G)≠1.
Relaxing the condition somewhat, and taking the subgroup of elements of a general finite group which centralize every chief factor, one simply gets the Fitting subgroup again (Huppert 1967, Kap.VI, Satz 5.4, p.686): The generalization to p-nilpotent groups is similar.
Consider the problem of trying to identify a normal subgroup H of G that contains its own centralizer and the Fitting group.
A group is said to be quasi-nilpotent if every element acts as an inner automorphism on every chief factor.
The generalized Fitting subgroup is the unique largest subnormal quasi-nilpotent subgroup, and is equal to the set of all elements which act as inner automorphisms on every chief factor of the whole group (Huppert & Blackburn 1982, Chapter X, Theorem 5.4, p. 126): Here an element g is in HCG(H/K) if and only if there is some h in H such that for every x in H, xg ≡ xh mod K. If G is a finite solvable group, then the Fitting subgroup contains its own centralizer.
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).
It is especially effective in the exceptional cases where components or signalizer functors are not applicable.