PART I, CHAPTER G
GENERAL GROUP THEORY
This book contains a single chapter in which we present most of the basic
group-theoretic material required for the proof of the Classification Theorem. As
noted in the Overview
[Iijpp.4-5] we shall refer to this chapter as [IG].
Moreover, a few topics will be omitted here and deferred to the parts of the proof
in which they are used.
Our Background References are listed on page 203; these contain proofs for a
substantial part of the theory of finite groups which we shall treat in this book.
Our list here is restricted to those Background References from the full list [Ii]
which we actually cite in the current chapter. As discussed in [Ii;§D], these are the
only references which we permit ourselves to quote to substantiate our arguments.
Thus the remainder of our references to the literature are to the separately listed
Expository References and are inessential to our logic.
In general, results in this chapter which are established in Background Ref-
erences will be limited to their statements, with appropriate citations. However,
we have made a couple of exceptions to this rule in the case of results of funda-
mental importance. For example, we develop the theory of the generalized Fitting
subgroup and of L-balance fully in Sections 3-5.
The vast majority of references to Background Results are to the general texts
[Gl], [Hul], [HuBl], [Sul] and [Al], with a limited number to [GL1], [Isl] and
[Fl]. In addition, a few major theorems are included in the Background Results
[Ii;§18]. Foremost among these is the Feit-Thompson Theorem, the solvability of
groups of odd order [FTl,BeGll], which we use without further reference. As a
starting point, we assume without reference the elements of finite group theory —
the basic definitions and results. Thus we assume familiarity with the definitions
of nilpotent and 7r-solvable groups, the (nilpotency) class of a group, the exponent
of a group, normal subgroup, composition series, composition factor, chief series,
chief factor, double coset, etc. Likewise we assume the basic properties of nilpotent
groups, commutators, Sylow's theorem, etc. All of this material can be found in the
Background References, particularly in [Gl;Chaps. 1-6]. Many further important
results from the basic texts will be restated here.
The chapter is divided into a considerable number of sections, reflecting the
various topics needed for the analysis. Most of these concern standard material of
local group theory: p-groups, transfer, elementary properties of strongly embedded
The first book in this series contained two chapters, an Overview [Ii] and an Outline of Proof
[I2]. The chapter following the current one shall cover properties of almost simple 3C-groups, and
be referred to as [1^]. In general, Chapter k of part I, II, etc. is denoted by [Ife], [life], etc.