2

PART I, CHAPTER G: GENERAL GROUP THEORY

subgroups, signalizer functors, various types of balance, factorization theorems,

etc. These results are largely of a general nature, but a substantial number involve

specialized assumptions dictated by the context in which they are to be applied.

Moreover, though in most cases our results hold for an arbitrary finite group X,

observations about the known simple groups cast a shadow over a noticeable portion

of this chapter in the following sense. First, some vital results actually require

special assumptions on suitable subgroups of X, assumptions which are valid if

these subgroups are DC-groups, that is, if their composition factors are known simple

groups. We discuss these assumptions in Section 2. Second, other results are

designed to apply to DC-groups and so their hypotheses, or the terms in which

they are formulated, include properties of certain DC-groups (to be established in

subsequent chapters). An example of the latter is the definition of semirigidity

related to the existence of terminal components (Section 7).

1. Notation

Throughout this series of monographs we reserve the letter G for the ambient

simple group under investigation — a DC-proper simple group [Ii;p.l2]. The DC-

proper hypothesis will not be used in this chapter except in the explicit way just

mentioned.

By contrast, X will denote an arbitrary finite group, and much of the material

to be discussed here will be stated for arbitrary X. On the other hand, when

discussing specific techniques — the Bender Method, for instance — it is preferable

to work with G itself. We shall therefore let the particular context determine which

of X or G to use in a given section.

In general, our notation is standard [Gl] and includes the bar convention for

homomorphic images—i.e., for any homomorphic image X, X, X, etc. of X, the

symbols Y, Y, Y, etc. will denote the corresponding image of any subgroup, sub-

set, or element 7 of X. Furthermore, unless otherwise explicitly stated, all

groups considered are finite. The discussion of amalgams in Sections 28 and

29, however, necessarily involves some infinite groups.

For clarity we repeat some important definitions and notations. As usual, we

write Y X if Y is a subgroup of X, Y X if Y is a proper subgroup of X, and

Y X if Y is a normal subgroup of X. Also X # is the set of non-identity elements

of X. For subsets which may not be subgroups we use the inclusion symbol C. If

A B X, we call B/A a section of X and denote by CX{B/A) and NX(B/A)

the preimages in N = NX{A) of C^{B) and Njj(B) respectively, where N = N/A.

The subgroup Y of X covers B/A if and only if B — (BnY)A. A group isomorphic

to a section of X is said to be involved in X.

Likewise, as usual, we write 7r(X) for the set of prime divisors of |X|, Z(X) for

the center of X, X' for the derived group [X, X] of X, and $(X) for the Frattini

subgroup of X. The subgroups Sol(X) and Sol7r(X) are the solvable and 7r-solvable

radicals of X (the unique maximal normal solvable and 7r-solvable subgroups of X,

respectively). The terms of the lower central series of X are X\ i = 1, 2,... , with

X 1 = X, X 2 = [X,X]; the terms of the upper central series of X are Z\{X) =

Z(X), ^ ( X ) , . . . ; and the terms of the derived series of X are X^\ i = 0 , 1 , . . . ,

starting with X° = X and X^ = X' and culminating in X(°°. A group X is

perfect if and only if X = [X, X], and is p-closed if and only if it has a unique Sylow

p-subgroup. The important subgroups F(X), E(X) and F*(X) will be defined in