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University of Delaware: Math 845. Fall 2011. F. Lazebnik

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University of Delaware: Math 845. Fall 2011. F. Lazebnik University of Delaware: Math 845. Fall 2011. F. Lazebnik. The first four lectures are suggested as a review for Math 845, Fall 2011. All references to numbered Sections and Exercises are for D.S. Dummit and R.M. Foote “Abstract Algebra”, 3rd edition, John Willey & Sons, Inc. These four lectures and suggested exercises are NOT mandatory, and can be easily replaced by introductory chapters to group theory from other undergraduate texts. Also in these notes you will see additional questions, some easy, some not–too–easy, and some very hard, often without indication of their difficulty. They are just to warm you up before the course, and to encourage you to learn to ask questions when you study a new topic. Because the number of axioms for a group is very small, the variety of groups is very large. As fewer axioms define groups than other basic structures, like rings, fields, modules, algebras, some people get an impression that groups are easier to study compared to those structures. I do not think this is true. I often feel much more restricted in tools when I think about abstract groups compared to, say, fields. The notion of a group is fundamental. Groups provide a mathematical language to discuss symmetry, when it is understood in the broadest sense. 1 2 Mathematics is just a tale about groups. – H. Poincar´e,1881. 3 Lecture 1. Review. A binary operation φ on a set A is a function from A×A to A, i.e., φ : A×A → A. φ((x, y)) is also denoted by xφy, or x · y, or just xy. A group (G, φ) is a non-empty set G with a binary operation φ, denoted φ(a, b) = ab, which is (i) Associative:(ab)c = a(bc) for all a, b, c ∈ G. (ii) There exists an identity element e = eG such that ea = ae = a for all a ∈ G ( e “two sided”). (iii) For every a ∈ G, there exists b ∈ G such that ab = ba = e. This element b is called an inverse of a in G, and is denoted by a−1.(a−1 is “two sided”). |X| will denote cardinality of a set X. For a group G, |G| is called the order of G. We write |G| = ∞, if G is infinite. G is trivial if G = {e}. G is abelian or commutative if ab = ba for all a, b ∈ G. Often the operation in an abelian group is denoted by +, and we write a + b instead of ab. In this case a−1 is denoted by −a. FACTS: In any group G the following properties hold. 0 (i) eG is unique. Indeed, if e is a possible another identity element, then ee0 = e as e0 is an identity, and ee0 = e0 as e is an identity. So e = e0. (ii) for each a ∈ G, the inverse of a is unique . Indeed, let b, b0 be inverses of a. Then consider the element (ba)b0. Since ba = e,(ba)b0 = eb0 = b0. Similarly, consider b(ab0). Since ab0 = e, b(ab0) = be = b. Due to the associativity, (ba)b0 = b(ab0). So b0 = b. (iii) (a−1)−1 = a. Indeed, aa−1 = a−1a = e by definition of a−1. Hence a = (a−1)−1, again by definition of an inverse of a−1. (iv) (ab)−1 = b−1a−1. Indeed, (ab)(b−1a−1) = a(bb−1)a−1 = a(e)a−1 = aa−1 = e. Similarly, (b−1a−1)(ab) = e. So b−1a−1 is the inverse of ab by the definition of the inverse. (v) Cancelation Law: in a group, ab = ac or ba = ca if and only if b = c. To see it, just multiply both sides by a−1 from the left in the first case, and from the right – in the second. (vi) The product a1·a2 ··· an, n ≥ 3, is understood as ((... (a1·a2)·a3) ...) ··· an, i.e., each time we apply it to two elements only. It can be shown to be independent on the way the factors are bracketed. For example, abcd = ((ab)c)d = (ab)(cd) = (a(bc))d = a(b(cd)). To show this, use induction on n. By definition, for every a ∈ G, a0 := e, a1 := a, and for an integer n ≥ 2, an := aaa ··· a, where we have n elements a and (n − 1 multiplications. Define a−n := (an)−1. It is easy to check that the exponents behave as they should: aman = am+n and (am)n = amn for all m, n ∈ Z. H is a subgroup of a group G = (G, φ), denoted H ≤ G, if ∅ 6= H ⊆ G and (H, φ|H ) is a group. Hence, the operation on H is the restriction of the operation of G on elements of H. It is easy to see that eG ∈ H, and eH = eG. hei and G itself are always subgroups of G. hei is called the trivial subgroup. If H ≤ G and H 6= G, then H is called a proper subgroup of G. To stress that H is a proper subgroup of G, we write H < G. 4 Proposition 1. H ≤ G if and only if H is closed under the operation of G and under taking inverses. Moreover, if G is finite, H ≤ G if and only if H is closed under the operation of G. If H ≤ G, then eH = eG and inverse of each a ∈ H in H is the inverse of a in G. Examples. • Every ring (in particularly, field) R is an abelian group under addition and the set of all units R× of ring R with 1 is a multiplicative group. In particularly, if F is a field, then F× = F − {0} is an abelian group under multiplication. nZ is defined as {nz : z ∈ Z}, and is a group under addition. Here is an example of two chains of groups. Groups in the first chain are under addition in C, and in the second chain – under multiplication in C. nZ < Z < Q < R < C; × × × × Z = {1, −1} < Q < R < C . • Every vector space is an abelian group under addition. • Let GL(n, F) denote the group of all invertible n × n matrices over a field F, or, equivalently, the group of all isomorphisms of an n-dimensional vector space over F. It is called the general linear group. The operation is the matrix multiplication, or, composition of isomorphisms, respectively. • SL(n, R) – the group of n × n matrices over an integral domain R1 with deter- minant 1. We have SL(n, R) ≤ GL(n, R). • S(X) – the group of all bijections of a set X to itself (the operation is composition of functions). It is called the symmetric group on X. Its elements are often referred to as permutations of X. For an infinite set X, the set of those bijections in S(X) which move (i.e, do not fix) only finitely many elements of X form a subgroup in S(X) denoted by Sfin(X). • Isom (E2) – the group of all distance preserving bijections (isometries) of points of the Euclidean plane E2. • Let A be a nonempty subset of a group G and let n1 n2 nk hAi = {a1 · a2 ··· ak : for all k ∈ Z, k ≥ 1, all ai ∈ A, and all ni ∈ Z} = n1 n2 nk {a1 · a2 ··· ak : for all k ∈ Z, k ≥ 1, all ai ∈ A, and all ni ∈ {−1, 1}. } Then hAi ≤ G. Moreover, hAi is the smallest (with respect to inclusion) subgroup of G containing A. If G = hAi, then it is said that A generates G. If G = hAi for a finite subset A, then G is called finitely generated. For g ∈ G, the subgroup h{g}i, or simply hgi, is called the cyclic subgroup generated by g. For g ∈ G, the order of hgi, is called the order of an element g, and is denoted by |g|. Hence the order of an element of a group is the smallest positive integer n such that gn = e, if such n exist. If it does not, we say that g has infinite order, and write |g| = ∞. If G = hgi, group G is called cyclic, and g is called a generator of G. 1 If you do not know the definition of an integral domain, think about it as of Z, or a field F, or as the polynomial ring of one or more indeterminants with coefficients from Z or F. 5 If G = hgi is a finite cyclic group of order n > 1, then G = {g, g2, . , gn−1, gn = e}, and all gi, 1 ≤ i ≤ n, are distinct. Also ga = gr if and only if a ≡ r (mod n). If G is an infinite cyclic group, then G = {gn, n ∈ Z}. Theorem 1. Let G = hgi, n ∈ N, and (a, n) = gcd(a, n). Then (i) If |g| = n, then |ga| = |hgai| = n/(a, n). Moreover, G = hgai if and only if (a, n) = 1. G has exactly exactly φ(n) generators, where φ is the Euler’s φ-function. (ii) If |g| = ∞, then ga is a generator of G if and only if a = ±1. (iii) Every subgroup H of G is cyclic. H generated by ga, where a is the smallest positive integer such that ga ∈ H. (iv) If |g| = n, and H = hgai ≤ G, then a divides n. Conversely, for every positive divisor d of n, there is a unique subgroup of G of order d. This subgroup is generated by gn/d. Proof. Prove all statements of this theorem.
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