Pages: 162

No chapter abstract available.

Pages: 21

In this chapter we present a collection of topics which serve as a background for the ideas you will see in this text. In Section 1.1, we present equivalence relations which foreshadow many important structures in abstract algebra such as cosets of a group, the orbits of a group action, and much more. In Section 1.2, we present basic ideas about functions, since all the structures presented in this text will have special functions associated with them. Section 1.3 will highlight some fundamental properties of the integers as a prelude to the more general topic of rings. In Section 1.4, we describe in more detail congruence modulo n which is an equivalence relation with additional properties. The equivalence classes of this relation produce one of the basic examples of a group and of a ring. Although, not explicitly presented, ideas in basic set theory are prevalent in this text and are an important background topic.

Pages: 65

In this chapter we introduce the reader to a group structure. In Section 2.1, we present basic definitions, examples and terminology related to groups. In Section 2.2, we define a subgroup, give many classic examples and present a shortcut for verifying subgroup. In Section 2.3, we present an important class of groups called cyclic groups. Permutation groups are an essential part of group theory and are presented in Section 2.4. In Section 2.5, we create new groups and subgroups by means of a product. In Section 2.6, we introduce functions between groups called homomorphisms. As a follow up, in Section 2.7, we use isomorphisms to define what it means for groups to be essentially equal (or isomorphic). In Section 2.8, we define perhaps one of the most equivalence relations on a group whose equivalence classes are called cosets, and these cosets can sometimes form a group. In Section 2.9, we investigate exactly when cosets form a group and look at several important examples. In Section 2.10, we investigate further notmal subgroups and consider groups which have no normal subgroups. Finally, in Section 2.11, we prove some fundamental isomorphism results for groups and factor groups.

Pages: 13

In this chapter we shall look at two families of non-abelian groups: The alternating group and the projective linear group. We will prove that they are both families of simple groups.

Pages: 34

In this chapter we present a powerful idea in group theory which is group action. The consequences of group action are many and the applications are incredibly useful as we shall see in the later sections in this chapter. Indeed, if someone wants to see concrete applications in group theory, then group action certainly fits the bill. In Section 4.1 we introduce the notion of group action. In Sections 4.2 and 4.3 we see a very nice application of group action in the field of combinatorics. Section 4.4 presents some deep theoretical consequences which follow from group action. The important Sylow Theorems in Section 4.5 follow from group action. We then use the Sylow Theorems to classify finite groups of particular sizes in Section 4.6, which in turn assists us in classifying completely finite abelian groups in Section 4.7.

Pages: 12

In this chapter we introduce two topics in group theory which can each fill an entire textbook. Therefore, one can view this chapter an giving exposure to this two topics with a light overview. In Section 5.1 we introduce the notion of a free group, which is a group generated by a set of elements, as a prelude to Section 5.2, group presentations, in which the group generators have additional rules for how they relate to each other. The free groups are free of these additional rules. Finally, in Section 5.3, we introduce group representations, in which we represent group elements by matrices.

Pages: 15

In this chapter we introduce two families of groups: nilpotent and solvable. One can think of these groups as generalizations of abelian groups. The main reason we cover this theory is for the application of solvable groups to answer the question if there exists a formula for finding the roots of a polynomial of degree five or more, otherwise known as solvability by radicals. Solvable groups are named as such because of this connection. In Section 6.1 we remind the reader of some important subgroups as well as introduce some additional important subgroups. In Section 6.2 we discuss certain chains of subgroups of a group which we use in Section 6.3 to define nilpotent and solvable groups.

Pages: 122

No chapter abstract available.

Pages: 31

In this chapter we now study a set with two operations as opposed to groups which have one operation. This, of course, makes the study both more complicated and more rich not only because of the second operation added, but also because of the interaction between the two operations. As we shall see, rings have many structures and concepts similar to groups such as subring, ring homomorphism and quotient ring and ideals (as opposed to normal subgroups).

Pages: 20

In this chapter we further explore properties of the integers and by doing so we define some special cases of integral domains. In Section 8.1, we introduce two of these special integral domains called Euclidean domain (ED) and principal ideal domain (PID). In Section 8.2, we introduce the third special integral domain called a unique factorization domain (UFD). In Section 8.3, we look at one particular integral domain which fails to be any of these three special integral domains. One area of study already discussed was to determine which property of a ring R carries over to the corresponding polynomial ring R[x]. In Section 8.4, we prove that the UFD property indeed carries over to the polynomial ring.

Pages: 30

In this chapter we set the stage for the next chapter on Galois theory. In Section 9.1, we remind the reader of some definitions, concepts and results as well as introduce the definition of algebraic. In Section 9.2, we investigate field extensions which allows us to look at chains of fields, an important idea in Galois theory. This also allows us to investigate some famous geometric impossibilities in Section 9.3. In Section 9.4, we look at some particular and important field extensions, and in Section 9.5, we prove the existence of some of these structures we introduced. Finally, in Section 9.6, we completely classify finite fields.

Pages: 39

In this chapter we make a big connection between field theory and the theory of groups called Galois theory. Such connections are ways in which powerful tools can be created for proving difficult mathematical results. The result which motivated Evariste Galois to develop this theory and its connections is the investigation of solvability by radicals. For the reader who is interested in the history of mathematics, Galois' biography is quite interesting and dramatic. In Section 10.1 we relate fields and groups via field homomorphisms of an extension field which fix the base field, thus arriving at Galois groups. We need Section 10.2 so that we can compute these Galois groups in a practical way. Section 10.3 is a bit off the beaten path, but at the very least the reader should be aware of the important results arrived at in this section. In Section 10.4 we introduce the notion of a splitting field and prove two important results linking the Galois group and the corresponding field extension. In Section 10.5 we introduce separable degree and link this concept to the size of the corresponding Galois group. In Section 10.6 introduce the notion of a Galois extension as we compare the lattice of subgroups of the Galois group and the corresponding lattice of intermediate fields in the corresponding field extension. In Section 10.7 we give the promised proof of Theorem 10.7 presented in Section 10.6 as well as prove a foundational result of Artin. In Section 10.8 we summarize our investigation in this chapter and prove the Fundamental Theorem of Galois Theory. In Section 10.9, we review Chapter 6 emphasizing the important concepts and results given therein regarding solvable groups. Finally, in Section 10.10, we investigate the notion of solvability by radicals.