Fitting Subgroup - The Generalized Fitting Subgroup

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.

The generalized Fitting subgroup can also be viewed as a generalized centralizer of chief factors. A nonabelian semisimple group cannot centralize itself, but it does act one itself as inner automorphisms. 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 1967, Kap.VI, Satz 5.4, p. 686):

Here an element g is in HC_G(H/K) if and only if there is some h in H such that for every x in H, xg ≡ xh mod K.