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Groups and Rings Groups and Rings David Pierce February , , : p.m. Matematik Bölümü Mimar Sinan Güzel Sanatlar Üniversitesi [email protected] http://mat.msgsu.edu.tr/~dpierce/ Groups and Rings This work is licensed under the Creative Commons Attribution–Noncommercial–Share-Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ CC BY: David Pierce $\ C Mathematics Department Mimar Sinan Fine Arts University Istanbul, Turkey http://mat.msgsu.edu.tr/~dpierce/ [email protected] Preface There have been several versions of the present text. The first draft was my record of the first semester of the gradu- ate course in algebra given at Middle East Technical University in Ankara in –. I had taught the same course also in –. The main reference for the course was Hungerford’s Algebra []. I revised my notes when teaching algebra a third time, in – . Here I started making some attempt to indicate how theorems were going to be used later. What is now §. (the development of the natural numbers from the Peano Axioms) was originally pre- pared for a course called Non-Standard Analysis, given at the Nesin Mathematics Village, Şirince, in the summer of . I built up the foundational Chapter around this section. Another revision, but only partial, came in preparation for a course at Mimar Sinan Fine Arts University in Istanbul in –. I expanded Chapter , out of a desire to give some indication of how mathematics, and especially algebra, could be built up from some simple axioms about the relation of membership—that is, from set theory. This building up, however, is not part of the course proper. The present version of the notes represents a more thorough-going revision, made during and after the course at Mimar Sinan. I try to make more use of examples, introducing them as early as possible. The number theory that has always been in the background has been integrated more explicitly into the text (see page ). I have tried to distinguish more clearly between what is essential to the course and what is not; the starred sections comprise most of what is not essential. All along, I have treated groups, not merely as structures satisfying certain axioms, but as structures isomorphic to groups of symme- tries of sets. The equivalence of the two points of view has been established in the theorem named for Cayley (in §., on page ). Now it is pointed out (in that section) that standard structures like 1 (Q+, 1, − , ) and (Q, 0, , +), are also groups, even though they are not obviously· symmetry− groups. Several of these structures are constructed in Chapter . (In earlier editions they were constructed later.) Symmetry groups as such are investigated more thoroughly now, in §§. and ., before the group axioms are simplified in §.. Rings are defined in Part I, on groups, so that their groups of units are available as examples of groups, especially in §. on semidirect products (page ). Also rings are needed to produce rings of matrices and their groups of units, as in §. (page ). I give many page-number references, first of all for my own conve- nience in the composition of the text at the computer. Thus the capabilities of Leslie Lamport’s LTEX program in automating such references are invaluable. Writing the text could hardly have been contemplated in the first place without Donald Knuth’s original TEX program. I now use the scrbook document class of KOMA-Script, “developed by Markus Kohm and based on earlier work by Frank Neukam” [, p. ]. Ideally every theorem would have an historical reference. This is a distant goal, but I have made some moves in this direction. The only exercises in the text are the theorems whose proofs are not already supplied. Ideally more exercises would be supplied, perhaps in the same manner. Contents Introduction . Mathematical foundations .. Setsandgeometry . .. Settheory........................ ... Notation . ... Classes and equality . ... Construction of sets . .. Functions and relations . .. An axiomatic development of the natural numbers . .. A construction of the natural numbers . .. Structures........................ .. Constructions of the integers and rationals . .. A construction of the reals . .. Countability . I. Groups . Basic properties of groups and rings .. Groups.......................... .. Symmetrygroups. ... Automorphism groups . ... Automorphism groups of graphs . ... A homomorphism . ... Cycles...................... Contents ... Notation . ... Even and odd permutations . .. Monoids and semigroups . ... Definitions ................... ... Some homomorphisms . ... Pi and Sigma notation . ... Alternating groups . .. Simplifications . ..... ..... ..... ..... .. Associativerings . . Groups .. *General linear groups . ... Additive groups of matrices . ... Multiplication of matrices . ... Determinants of matrices . ... Inversion of matrices . ... Modules and vector-spaces . .. New groups from old . ... Products .................... ... Quotients . ... Subgroups. ... Generated subgroups . .. Order .......................... .. Cosets .......................... .. Lagrange’s Theorem . .. Normalsubgroups . .. Classification of finite simple groups . ... Classification . ... Finite simple groups . . Category theory .. Products......................... .. Sums........................... Contents .. *Weakdirectproducts . .. Freegroups ....................... .. *Categories . ... Products .................... ... Coproducts . ... Free objects . .. Presentation of groups . .. Finitely generated abelian groups . . Finite groups .. Semidirectproducts . .. Cauchy’s Theorem . .. Actions of groups . ... Centralizers . ... Normalizers . ... Sylow subgroups . .. *Classification of small groups . .. Nilpotent groups . .. Solublegroups . .. Normalseries ...................... II. Rings . Rings .. Rings .......................... .. Examples ........................ .. Associative rings . .. Ideals .......................... . Commutative rings .. Commutative rings . .. Division ......................... Contents .. *Quadratic integers . .. Integral domains . .. Localization. .. *Ultraproducts of fields . ... Zorn’s Lemma . ... Boolean rings . ... Regular rings . ... Ultraproducts . .. Polynomial rings . ... Universal property . ... Division . ... *Multiple zeros . ... Factorization . A. The German script Bibliography List of Figures .. Acycle. ......................... .. TheButterflyLemma . A.. TheGermanalphabet . Introduction Published around b.c.e., the Elements of Euclid is a model of mathematical exposition. Each of its thirteen books consists mainly of statements followed by proofs. The statements are usually called Propositions today [, ], although they have no particular title in the original text []. By their content, they can be understood as theorems or problems. Writing six hundred years after Euclid, Pappus of Alexandria explains the difference [, p. ]: Those who wish to make more skilful distinctions in geometry find it worthwhile to call • a problem (πρόβλημα), that in which it is proposed (προβάλλεται) to do or construct something; • a theorem (θεώρημα), that in which the consequences and necessary implications of certain hypotheses are investigated (θεωρεῖται). For example, Euclid’s first proposition is the the problem of con- structing an equilateral triangle. His fifth proposition is the theo- rem that the base angles of an isosceles triangle are equal to one another. Each proposition of the present notes has one of four titles: Lemma, Theorem, Corollary, or Porism. Each proposition may be fol- lowed by an explicitly labelled proof, which is terminated with a box . If there is no proof, the reader is expected to supply her or his own proof, as an exercise. No propositions are to be accepted on faith. Nonetheless, for an algebra course, some propositions are more im- portant than others. The full development of the foundational Chap- ter below would take a course in itself, but is not required for algebra as such. In these notes, a proposition may be called a lemma if it will be used to prove a theorem, but then never used again. Lemmas in these notes are numbered sequentially. Theorems are also numbered sequentially, independently from the lemmas. A statement that can be proved easily from a theorem is called a corollary and is numbered with the theorem. So for example Theorem on page is followed by Corollary .. Some propositionss can be obtained easily, not from a preceding the- orem itself, but from its proof. Such propositions are called porisms and, like corollaries, are numbered with the theorems from whose proofs they are derived. So for example Porism . on page follows Theorem . The word porism and its meaning are explained, in the th century c.e., by Proclus in his commentary on the first book of Euclid’s Elements [, p. ]: “Porism” is a term applied to a certain kind of problem, such as those in the Porisms of Euclid. But it is used in its special sense when as a result of what is demonstrated some other theorem comes to light without our propounding it. Such a theorem is therefore called a “porism,” as being a kind of incidental gain re- sulting from the scientific demonstration. The translator explains that the word porism comes from the verb πορίζω, meaning to furnish or provide. The original source for much of the material of these notes is Hunger- ford’s Algebra [], or sometimes Lang’s Algebra [], but there are various rearrangements and additions. The back cover
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