Fitting subgroup

In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which 'controls' the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroup F*, which is generated by the Fitting subgroup and the components of G. In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which 'controls' the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroup F*, which is generated by the Fitting subgroup and the components of G. For an arbitrary (not necessarily finite) group G, the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of G. For infinite groups, the Fitting subgroup is not always nilpotent. The remainder of this article deals exclusively with finite groups. The nilpotency of the Fitting subgroup of a finite group is guaranteed by Fitting's theorem which says that the product of a finite collection of normal nilpotent subgroups of G is again a normal nilpotent subgroup. 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. Similarly the Fitting subgroup of G/F(G) will be nontrivial if G is not itself nilpotent, giving rise to the concept of Fitting length. Since the Fitting subgroup of a finite solvable group contains its own centralizer, this gives a method of understanding finite solvable groups as extensions of nilpotent groups by faithful automorphism groups of nilpotent groups. In a nilpotent group, every chief factor is centralized by every element. 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. A component of a group is a subnormal quasisimple subgroup. (A group is quasisimple if it is a perfect central extension of a simple group.) The layer E(G) or L(G) of a group is the subgroup generated by all components. Any two components of a group commute, so the layer is a perfect central extension of a product of simple groups, and is the largest normal subgroup of G with this structure. The generalized Fitting subgroup F*(G) is the subgroup generated by the layer and the Fitting subgroup. The layer commutes with the Fitting subgroup, so the generalized Fitting subgroup is a central extension of a product of p-groups and simple groups. The layer is also the maximal normal semisimple subgroup, where a group is called semisimple if it is a perfect central extension of a product of simple groups.