Fitting subgroup

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In group theory, a branch of mathematics, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the 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.

1 The Fitting subgroup
2 The generalized Fitting subgroup
3 Properties
4 Applications
5 Further reading

[edit] The Fitting subgroup

The existence of the Fitting subgroup is guaranteed by Fitting's theorem which says that the product of a 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 solvable group (not equal to the trivial group), then the Fitting subgroup is always non-trivial, i.e. F(G)≠ 1 (compare with the statement below that the Fitting subgroup contains its own centralizer in G).

[edit] The generalized Fitting subgroup

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.

The definition of the generalized Fitting subgroup looks a little strange at first. To motivate it, consider the problem of trying to find a normal subgroup H of G that contains its own centralizer and the Fitting group. If C is the centralizer of H we want to prove that C is contained in H. If not, pick a minimal characteristic subgroup M/Z(H) of C/Z(H), where Z(H) is the center of H, which is the same as the intersection of C and H. Then M/Z(H) is a product of simple or cyclic groups as it is characteristically simple. If M/Z(H) is a product of cyclic groups then M must be in the Fitting subgroup. If M/Z(H) is a product of non-abelian simple groups then the derived subgroup of M is a normal semisimple subgroup mapping onto M/Z(H). So if H contains the Fitting subgroup and all normal semisimple subgroups, then M/Z(H) must be trivial, so H contains its own centralizer. The generalized Fitting subgroup is the smallest subgroup that contains the Fitting subgroup and all normal semisimple subgroups.

[edit] Properties

If G is solvable, then the Fitting subgroup contains its own centralizer, and is the same as the generalized Fitting subgroup. If G is any (finite) group, 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.

[edit] Applications

A 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 what field a simple group should be defined over. (But 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 or 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.