11 - The generalized Fitting subgroup

Summary

We've seen that the composition factors of a finite group control the structure of the group in part, but that control is far from complete. Section 31 introduces a tool for studying finite groups via composition factors ‘near the bottom’ of the group. The generalized Fitting subgroup F^*(G) of a finite group G is a characteristic subgroup of G generated by the small normal subgroups of G and with the property that C_G(F^*(G)) ≤ F^*(G). This last property supplies a representation of G as a subgroup of Aut(F^*(G)) with kernel Z(F^*(G)). G can be effectively investigated via this representation because F^*(G) is a relatively uncomplicated group whose embedding in G is particularly well behaved.

It turns out that F^*(G) is a central product of the groups O_p(G), p ε π(G), with a subgroup E(G) of G. To define E(G) requires some terminology. A central extension of a group X is a group Y together with a surjective homomorphism of Y onto X whose kernel is in the center of Y. The group Y will also be said to be a central extension of X. A group L is quasisimple if L is perfect and the central extension of a simple group. The components of G are its subnormal quasisimple subgroups, and E(G) is the subgroup of G generated by the components of G. It develops that E(G) is a central product of the components of G.