Chapter IV. Groups and Group Actions

Abstract

This chapter develops the basics of group theory, with particular attention to the role of group actions of various kinds. The emphasis is on groups in Sections 1–3 and on group actions starting in Section 6. In between is a two-section digression that introduces rings, fields, vector spaces over general fields, and polynomial rings over commutative rings with identity.

Section 1 introduces groups and a number of examples, and it establishes some easy results. Most of the examples arise either from number-theoretic settings or from geometric situations in which some auxiliary space plays a role. The direct product of two groups is discussed briefly so that it can be used in a table of some groups of low order.

Section 2 defines coset spaces, normal subgroups, homomorphisms, quotient groups, and quotient mappings. Lagrange's Theorem is a simple but key result. Another simple but key result is the construction of a homomorphism with domain a quotient group $G/H$ when a given homomorphism is trivial on $H$. The section concludes with two standard isomorphism theorems.

Section 3 introduces general direct products of groups and direct sums of abelian groups, together with their concrete “external” versions and their universal mapping properties.

Sections 4–5 are a digression to define rings, fields, and ring homomorphisms, and to extend the theories concerning polynomials and vector spaces as presented in Chapters I–II. The immediate purpose of the digression is to make prime fields and the notion of characteristic available for the remainder of the chapter. The definitions of polynomials are extended to allow coefficients from any commutative ring with identity and to allow more than one indeterminate, and universal mapping properties for polynomial rings are proved.

Sections 6–7 introduce group actions. Section 6 gives some geometric examples beyond those in Section 1, it establishes a counting formula concerning orbits and isotropy subgroups, and it develops some structure theory of groups by examining specific group actions on the group and its coset spaces. Section 7 uses a group action by automorphisms to define the semidirect product of two groups. This construction, in combination with results from Sections 5–6, allows one to form several new finite groups of interest.

Section 8 defines simple groups, proves that alternating groups on five or more letters are simple, and then establishes the Jordan–Hölder Theorem concerning the consecutive quotients that arise from composition series.

Section 9 deals with finitely generated abelian groups. It is proved that “rank” is well defined for any finitely generated free abelian group, that a subgroup of a free abelian group of finite rank is always free abelian, and that any finitely generated abelian group is the direct sum of cyclic groups.

Section 10 returns to structure theory for finite groups. It begins with the Sylow Theorems, which produce subgroups of prime-power order, and it gives two sample applications. One of these classifies the groups of order $pq$, where $p$ and $q$ are distinct primes, and the other provides the information necessary to classify the groups of order 12.

Section 11 introduces the language of “categories” and “functors.” The notion of category is a precise version of what is sometimes called a “context” at points in the book before this section, and some of the “constructions” in the book are examples of “functors.” The section treats in this language the notions of “product” and “coproduct,” which are abstractions of “direct product” and “direct sum.”