Algebra. Travis Dirle. December 4, 2016

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1 Abstract Algebra

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3 Algebra Travis Dirle December 4, 2016

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5 Contents 1 Groups Semigroups, Monoids and Groups Homomorphisms and Subgroups Cyclic Groups Cosets and Counting Normality, Quotient Groups, and Homomorphisms Symmetric, Alternating, and Dihedral Groups Direct Products and Direct Sums The Structure of Groups Free Abelian Groups Finitely Generated Abelian Groups The Krull-Schmidt Theorem The Action of a Group on a Set The Sylow Theorems Classification of Finite Groups Nilpotent and Solvable Groups Normal and Subnormal Series Rings Rings and Homomorphisms Ideals Factorization in Commutative Rings Rings of Quotients and Localization Rings of Polynomials and Formal Power Series Factorization in Polynomial Rings Modules Modules, Homomorphisms and Exact Sequences Free Modules and Vector Spaces Projective and Injective Modules Hom and Duality Tensor Products Modules over a Principle Ideal Domain i

6 CONTENTS 4.7 Algebras Fields and Galois Theory Field Extensions The Fundemental Theorem Splitting Fields, Algebraic Closure and Normality The Galois Group of a Polynomial Finite Fields Separability Cyclic Extensions Cyclotomic Extensions Radical Extensions Commutative Rings and Modules Chain Conditions Prime and Primary Ideals Primary Decomposition Noetherian Rings and Modules Ring Extensions Dedekind Domains The Hilbert Nullstellensatz Linear Algebra Matrices and Maps Rank and Equivalence Determinants Decomposition of a Single Linear Transformation and Similarity The Characteristic Polynomial, Eigenvectors and Eigenvalues.. 87 ii

7 Chapter 1 Groups 1.1 SEMIGROUPS, MONOIDS AND GROUPS Definition If G is a nonempty set, a binary operation on G is a function G G G. A semigroup is a nonempty set G together with a binary operation on G which is (i) associative: a(bc) = (ab)c for all a, b, c G; A monoid is a semigroup G which contains a (ii) (two-sided) identity element e G such that ae = ea = a for all a G. A group is a monoid G such that (iii) for every a G there exists a (two-sided) inverse element a 1 G such that a 1 a = aa 1 = e. A semigroup G is said to be abelian or commutative if its binary operation is (iv) commutative: ab = ba for all a, b G. Theorem If G is a monoid, then the identity element e is unique. If G is a group, then (i) c G and cc = c c = e; (ii) for all a, b, c G ab = ac b = c and ba = ca b = c (left and right cancellation); (iii) for each a G, the inverse element a 1 is unique; (iv) for each a G, (a 1 ) 1 = a; (v) for a, b G, (ab) 1 = b 1 a 1 ; (vi) for a, b G the equations ax = b and ya = b have unique solutions in G: x = a 1 b and y = ba 1. Theorem Let R( ) be an equivalence relation on a monoid G such that a 1 a 2 and b 1 b 2 imply a 1 b 1 a 2 b 2 for all a i, b i G. Then the set G/R of all equivalence classes of G under R is a monoid under the binary operation defined by ā b = āb, where x denotes the equivalence class of x G. If G is an (abelian) group, then so is G/R. Definition Let G be a semigroup, a G and n N. The element a n G is defined to be the standard n product n i=1 a i with a i = a for 1 i n. If G

8 CHAPTER 1. GROUPS is a monoid, a 0 is defined to be the identity element e. If G is a group, then for each n N, a n is defined to be (a 1 ) n G. The order of a group G is the cardinal number G. Theorem If G is a group (resp. semigroup, monoid) and a G, then for all m, n Z: (i) a m a n = a m+n (additive notation: ma + na = (m + n)a); (ii) (a m ) n = a mn (additive notation: n(ma) = mna). If a group G is abelian, then (ab) n = a n b n for all n Z and all a, b G. 1.2 HOMOMORPHISMS AND SUBGROUPS Definition Let G and H be semigroups. A function f : G H is a homomorphism provided f(ab) = f(a)f(b), for all a, b G. (1.1) If f is injective as a map of sets, f is said to be a monomorphism. If f is surjective, f is called an epimorphism. If f is bijective, f is called an isomorphism. In this case G and H are said to be isomorphic (written G = H). A homomorphism f : G G is called an endomorphism of G and an isomorphism f : G G is called an automorphism of G. Definition Let f : G H be a homomorphism of groups. The kernel of f (denoted Ker f) is {a G : f(a) = e H}. If A is a subset of G, then f(a) = {b H : b = f(a) for some a A} is the image of A. f(g) is called the image of f and denoted Im f. If B is a subset of H, f 1 (B) = {a G : f(a) B} is the inverse image of B. Theorem Let f : G H be a homomorphism of groups. Then (i) f is a monomorphism if and only if Ker f = {e}; (ii) f is an isomorphism if and only if there is a homomorphism f 1 : H G such that ff 1 = 1 H and f 1 f = 1 G. Definition Let G be a group and H a nonempty subset that is closed under the product in G. If H is itself a group under the product in G, then H is said to be a subgroup of G. This is denoted H < G. Theorem Let H be a nonempty subset of a group G. Then H is a subgroup of G if and only if ab 1 H for all a, b H. Corollary If G is a group and {H i : i I} is a nonempty family of subgroups, then i I H i is a subgroup of G. 2

9 CHAPTER 1. GROUPS Definition Let G be a group and X a subset of G. Let {H i : i I be the family of all subgroups of G which contain X. Then i I H i is called the subgroup of G generated by the set X and denoted X. Theorem If G is a group and X is a nonempty subset of G, then the subgroup X generated by X consists of all finite products a n 1 1 a n 2 2 a nt t (a i X; n i Z). In particular for every a G, a = {a n : n Z}. Remark Composition of homomorphisms (resp. mono, epi, and isomorphisms) of semigroups, is also a homomorphism. For a homomorphism of groups f : G Hthen f(e G ) = e H. Furthermore f(a 1 ) = f(a) 1 for all a G. Two examples of subgroups of a group G are G itself and the trivial subgroup e. A subgroup H such that H G, H e is called a proper subgroup. If G = a 1,..., a n, (a i G), G is said to be finitely generated.if a G, the subgroup a is called the cyclic (sub)group generated by a. If H and K are subgroups, the subgroup H K generated by H and K is called the join of H and K and is denoted H K (additive notation: H + K). 1.3 CYCLIC GROUPS Theorem Every subgroup H of the additive group Z is cyclic. Either H = 0 or H = m, where m is the least positive integer in H. If H 0, then H is infinite. Theorem Every infinite cyclic group is isomorphic to the additive group Z and every finite cyclic group of order m is isomorphic to the additive group Z m. Definition Let G be a group and a G. The order of a is the order of the cyclic subgroup a and is denoted a. Theorem Let G be a group and a G. If a has infinite order, then (i) a k = e if and only if k = 0; (ii) the elements a k (k Z) are all distinct. If a has finite order m > 0, then (iii) m is the least positive integer such that a m = e; (iv) a k = e if and only if m k; (v) a r = a s if and only if r s( mod m); (vi) a consists of the distinct elements a, a 2,..., a m 1, a m = e; (vii) for each k such that k m, a k = m/k. Theorem Every homomorphic image and every subgroup of a cyclic group G is cyclic. In particular, if H is a nontrivial subgroup of G = a and m is the least positive integer such that a m H then H = a m. 3

10 CHAPTER 1. GROUPS Theorem Let G = a be a cyclic group. If G is infinite, then a and a 1 are the only generators of G. If G is finite of order m, then a k is a generator of G if and only if (k, m) = COSETS AND COUNTING Definition Let H be a subgroup of a group G and a, b G. a is right congruent to b modulo H, denoted a r b(modh), if ab 1 H. a is left congruent to b modulo H, denoted a l b(modh), if a 1 b H. Theorem Let H be a subgroup of a group G. (i) Right (resp. left) congruence modulo H is an equivalence relation on G. (ii) The equivalence class of a G under right (resp. left) congruence modulo H is the set Ha = {ha : h H} (resp. ah = {ah : h H}). (iii) Ha = H = ah for all a G. The set Ha is called a right coset of H in G and ah is called a left coset of H in G. In general it is not the case that a right coset is also a left coset. Corollary Let H be a subgroup of a group G. (i) G is the union of the right (resp. left) cosets of H in G. (ii) Two right (resp. left) cosets of H in G are either disjoint or equal. (iii) For all a, b G, Ha = Hb ab 1 H and ah = bh a 1 b H. (iv) If R is the set of distinct right cosets of H in G and L is the set of distinct left cosets of H in G, then R = L. Definition Let H be a subgroup of a group G. The index of H in G, denoted [G : H], is the cardinal number of the set of distinct right (resp. left) cosets of H in G. Theorem If K,H,G are groups with K < H < G, then [G : K] = [G : H][H : K]. If any two of these indices are finite, then so is the third. Corollary (Lagrange). If H is a subgroup of a group G, then G = [G : H] H. In particular if G is finite, the order a of a G divides G. If G is a group and H,K are subsets of G, we denote by HK the set {ab : a H, b K}; If H,K are subgroups, HK may not by a subgroup. Theorem Let H and K be finite subgroups of a group G. Then HK = H K / H K. Proposition If H and K are subgroups of a group G, then [H : H K] [G : K]. If [G : K] is finite, then [H : H K] = [G : K] if and only if G = KH. 4

11 CHAPTER 1. GROUPS Proposition Let H and K be subgroups of finite index of a group G. Then [G : H K] is finite and [G : H K] [G : H][G : K]. Furthermore, [G : H K] = [G : H][G : K] if and only if G = HK. 1.5 NORMALITY, QUOTIENT GROUPS, AND HOMOMORPHISMS Theorem If N is a subgroup of a group G, then the following conditions are equivalent. (i) Left and right congruence modulo N coincide (that is, define the same equivalence relation on G); (ii) every left coset of N in G is a right coset of N in G; (iii) an = Na for all a G; (iv) for all a G, ana 1 N, where ana 1 = {ana 1 : n N}; (v) for all a G, ana 1 = N. Definition A subgroup N of a group G which satisfies the equivalent conditions is said to be normal in G (or a normal subgroup of G); we write N G if N is normal in G. Every subgroup of an abelian group is trivially normal. The intersection of a family of normal subgroups is a normal subgroup. If G is a group with subgroups N and M such that N M and M G, it does not follow that N G. Theorem Let K and N be subgroups of a group G with N normal in G. Then (i) N K is a normal subgroup of K; (ii) N is a normal subgroup of N K; (iii) NK = N K = KN; (iv) if K is normal in G and K N = e then nk = kn for all k K and n N. Theorem If N is a normal subgroup of a group G and G/N is the set of all (left) cosets of N in G, then G/N is a group of order [G : N] under the binary operation given by (an)(bn) = abn. If N is a normal subgroup of a group G, then the group G/N,is called the quotient group or factor group of G by N. If written additively, we have (a + N) + (b + N) = (a + b) + N. Theorem If f : G H is a homomorphism of groups, then the kernel of f is a normal subgroup of G. Conversely, if N is a normal subgroup of G, then the map π : G G/N given by π(a) = an is an epimorphism with kernel N. The map π : G G/N is called the canonical epimorphism or projection 5

12 CHAPTER 1. GROUPS Theorem If f : G H is a homomorphism of groups and N is a normal subgroup of G contained in the kernel of f, then there is a unique homomorphism f : G/N H such that f(an) = f(a) for all a G. Imf = Im f and Ker f = (Kerf)/N. f is an isomorphism if and only if f is an epimorphism and N = Kerf. Theorem (First Isomorphism Theorem) If f : G H is a homomorphism of groups, then f induces an isomorphism G/Kerf = Imf. Corollary If f : G H is a homomorphism of groups, N G, M H, and f(n) < M, then f induces a homomorphism f : G/N H/M, given by an f(a)m. f is an isomorphism if and only if Imf M = H and f 1 (M) N. In particular if f is an epimorphism such that f(n) = M and Kerf N, then f is an isomorphism. Corollary (Second Isomorphism Theorem) If K and N are subgroups of a group G, with N normal in G, then K/(N K) = NK/N. Corollary (Third Isomorphism Theorem) If H and K are normal subgroups of a group G such that K < H, then H/K is a normal subgroup of G/K and (G/K)/(H/K) = G/H. Theorem If f : G H is an epimorphism of groups, then the assignment K f(k) defines a one-to-one correspondence between the set S f (G) of all subgroups K of G which contain Ker f and the set S(H) of all subgroups of H. Normal subgroups correspond to normal subgroups. Corollary If N is a normal subgroup of a group G, then every subgroup G/N is of the form K/N where K is a subgroup of G that contains N. Furthermore, K/N is normal in G/N if and only if K is normal in G. 1.6 SYMMETRIC, ALTERNATING, AND DIHEDRAL GROUPS The symmetric group S n is the group of all bijections I n I n, where I n = {1, 2,..., n}. The elements of S n are called permutations. Definition Let i 1, i 2,..., i r, (r n) be distinct elements of I n = {1, 2,..., n}. Then (i 1 i 2 i 3... i r ) denotes the permutation that maps i 1 i 2, i 2 i 3,..., i r i 1, and maps every other element of I n onto itself. (i 1 i 2 i r ) is called a cycle of length r or an r-cycle; a 2-cycle is called a transposition. A 1-cycle (k) is the identity permutation. Clearly, an r-cycle is an element of order r in S n. The inverse of the cycle (i 1 i 2 i r ) is the cycle (i r i r 1 i r 2 i 2 i 1 ) = (i 1 i r i r 1 i 2 ). 6

13 CHAPTER 1. GROUPS Definition The permutations σ 1, σ 2,..., σ r ofs n are said to be disjoint provided that for each 1 i r, and every k I n, σ i (k) k implies σ j (k) = k for all j i. In other words σ 1, σ 2,..., σ r are disjoint if and only if no element of I n is moved by more than one of them. στ = τσ whenever σ and τ are disjoint. Theorem Every nonidentity permutation in S n is uniquely (up to the order of the factors) a product of disjoint cycles, each of which has length at least 2. Corollary The order of a permutation σ S n is the least common multiple of the orders of its disjoint cycles. Corollary Every permutation in S n can be written as a product of (not necessarily disjoint) transpositions. Definition A permutation τ S n is said to be even (resp. odd) if τ can be written as a product of an even (resp. odd) number of transpositons. Theorem A permutation in S n (n 2) cannot be both even and odd. Theorem For each n 2, let A n be the set of all even permutations of S n. Then A n is a normal subgroup of S n of index 2 and order S n /2 = n!/2. Furthermore A n is the only subgroup of S n of index 2. The group A n is called the alternating group on n letters Definition A group G is said to be simple if G has no proper normal subgroups. Theorem The alternating group A n is simple if and only if n 4. Recall that if τ is a 2-cycle, τ 2 = (1) and hence τ = τ 1. D n is called the dihedral group of degree n. It is isomorphic to and usually identified with the group of all symmetries of a regular polygon with n sides. Theorem For each n 3 the dihedral group D n is a group of order 2n whose generators a and b satisfy: (i) a n = (1), b 2 = (1), a k (1) if 0 < k < n; (ii) ba = a 1 b. 1.7 DIRECT PRODUCTS AND DIRECT SUMS Definition Let G 1, G 2..., G n be a finite collection of groups. The external direct product of G 1, G 2,..., G n, written as G 1 G 2 G n, is the set of all n-tuples for which the ith component is an element of G i and the operation is componentwise. In symbols, G 1 G 2 G n = {(g 1, g 2,..., g n ) : g i G i } 7

14 CHAPTER 1. GROUPS Theorem If {G i : i I} is a family of groups, then (i) the direct product i I G i is a group; (ii) for each k I the map π k : i I G i G k given by f f(k) (or {a i } a k ) is an epimorphism of groups. This is often called the canonical projection of the direct product. Proposition If G 1,..., G n are groups, their direct product is a group of order G 1 G 2 G n (if any G i is infinite, so is the direct product. Proposition Let G 1, G 2,..., G n be groups and let G = G 1 G n be their direct product. (i) For each fixed i the set of elements of G which have the identity of G j in the j th position for all j i and arbitrary elements of G i in position i is a subgroup of G isomorphic to G i : G i = {(1, 1,..., 1, gi, 1,..., 1) : g i G i }. If we identify G i with this subgoup, then G i G and G/G i = G1 G i 1 G i+1 G n. (ii) For each fixed i define π i : G G i by π i ((g 1, g 2,..., g n )) = g i. Then π i is a surjective homomorphism with kerπ i = {(g 1,..., g i 1, 1, g 1+1,..., g n ) : g j G j for all j i} = G 1 G i 1 G i+1 G n (here the 1 appears in position i). (iii) Under the identification in part (i), if x G i and y G j for some i j, then xy = yx. Corollary An external direct product G 1 G 2 G n of a finite number of finite cyclic groups is cyclic if and only if G i and G j are relatively prime when i j. Definition Let H 1, H 2,..., H n be a finite collection of normal subgroups of G. We say that G is the internal direct product of H 1, H 2,..., H n and write G = H 1 H 2 H n, if (i) G = H 1 H 2 H n = {h 1 h 2 h n : h i H i } (ii) (H 1 H 2 H i ) H i+1 = {e} for i = 1, 2,..., n 1. Theorem If a group G is the internal direct product of a finite number of subgroups H 1, H 2,..., H n, then G is isomorphic to the external direct product of H 1, H 2,..., H n. So H 1 H 2 H n = H1 H 2 H n. Theorem For {G i : i I} a family of groups, define w i I G i (called the weak direct product to be the set of f i I G i such that f(i) = e i, the identity in G i, for all but a finite number of i, then (i) w i I G i is a normal subgroup of i I G i; (ii) for each k I the map i k : G k w i I G i given by i k (a) = {a i } i I, where a i = e for i k and a k = a, is a monomorphism of groups; (iii) for each i I, i i (G i ) is a normal subgroup of i I G i. The maps i k are called the canonical injections. Theorem Let {N i : i I} be a family of normal subgroups of a group G such that 8

15 CHAPTER 1. GROUPS (i) G = i I N i ; (ii) for each k I, N k i k N i = e. Then G = w i I N i. Here N 1 N 2 N r = N 1 N 2 N r = {n 1 n 2 n r : n i N i }. Corollary If N 1, N 2,..., N r are normal subgroups of a group G such that G = N 1 N 2 N r and for each 1 k r, N k (N 1 N k 1 N k+1 N r ) = e, then G = N 1 N 2 N r. Proposition Let {G i : i I} and {N i : i I} be families of groups such that N i is a normal subgroup of G i for each i I. (i) i I N i is a normal subgroup of i I G i and i I G i / i I N i = i I G i/n i. (ii) w i I N i is a normal subgroup of w i I G i and w i I G i / w i I N i = w i I G i/n I. 9

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17 Chapter 2 The Structure of Groups 2.1 FREE ABELIAN GROUPS A basis of an abelian group F is a subset X of F such that (i) F = X ; and (ii) for distinct x 1, x 2,..., x k X and n i Z, n 1 x 1 +n 2 x 2 + +n k x k = 0 n i = 0 for every i. An abelian group F that satisfies the following condidtions is called a free abelian group (on the set X) Theorem The following conditions on an abelian group F are equivalent. (i) F has a nonempty basis. (ii) F is the internal direct sum of a family of infinite cyclic subgroups. (iii) F is isomorphic to a direct sum of copies of the additive group Z. (iv) There exists a nonempty set X and a function i : X F with the following property: given an abelian group G and function f : X G, there exists a unique homomorphism of groups f : F G such that fi = f. Theorem Any two bases of a free abelian group F have the same cardinality. The cardinal number of a basis X, X, is called the rank of F. Proposition Let F 1 be the free abelian group on the set X 1 and F 2 the free abelian group on the set X 2. Then F 1 = F2 if and only if F 1 and F 2 have the same rank. Theorem Every abelian group G is the homomorphic image of a free abelian group of rank X, where X is a set of generators of G. Theorem If F is a free abelian group of finite rank n and G is a nonzero subgroup of F, then there exists a basis {x 1,..., x n } of F, an integer r (1 r n) and positive integers d 1,..., d r such that d 1 d 2 d r and G is free abelian with basis {d 1 x 1,..., d r x r }. Corollary If G is a finitely generated abelian group generated by n elements, then every subgroup H of G may be generated by m elements with m n. 11

18 CHAPTER 2. THE STRUCTURE OF GROUPS 2.2 FINITELY GENERATED ABELIAN GROUPS Theorem Every finitely generated abelian group G is isomorphic to a finite direct sum of cyclic groups in which the finite cyclic summands (if any) are of orders m 1,..., m t, where m 1 > 1 and m 1 m 2 m t. Theorem Every finitely generated abelian group G is isomorphic to a finite direct sum of cyclic groups, each of which is either infinite or of order a power of a prime. Lemma If m is a positive integer and m = p n 1 1 p n 2 2 p nt t (p 1,..., p t dis-. tinct primes and each n i > 0), then Z m = Zp n 1 1 Z p n 2 2 We have that Z rn = Zr Z n whenever (r, n) = 1. Z p n t t Corollary If G is a finite abelian group of order n, then G has a subgroup of order m for every positive integer m that divides n. Lemma Let G be an abelian group, m an integer and p a prime integer. Then each of the following is a subgroup of G: (i) mg = {mu : u G}; (ii) G[m] = {u G : mu = 0}; (iii) G(p) = {u G : u = p n for some n 0}; (iv) G t = {u G : u is finite }. In particular there are isomorphisms (v) Z p n[p] = Z p (n 1) and p m Z p n = Zp n m(m < n). Let H and G i (i I) be abelian groups. (vi) If g : G i I G i is an isomorphism, then the restriction of g to mg and G[m] respectively are isomorphisms mg = i I mg i and G[m] = i I G i[m]. (vii) If f : G H is an isomorphism, then the restrictions of f to G t and G(p) respectively are isomorphisms G t = Ht and G(p) = H(p). If G is an abelian group, then the subgroup G t is called the torsion subgroup of G. If G = G t, then G is said to be a torsion group. If G t = 0 then G is said to be torsion-free. Theorem Let G be a finitely generated abelian group. (i) There is a unique nonnegative integer s such that the number of infinite cyclic summands in any decomposition of G as a direct sum of cyclic groups is precisely s; (ii) either G is free abelian or there is a unique list of (not necessarily distinct) positive integers m 1,..., m t such that m 1 > 1, m 1 m 2 m t and G = Z m1 Z mt F with F free abelian; 12

19 CHAPTER 2. THE STRUCTURE OF GROUPS (iii) either G is free abelian or there is a list of positive integers p s 1 1,..., p s k k, which is unique except for the order of its members, such that p 1,..., p k are (not necessarily distinct) primes, s 1,..., s k are (not necessarily distinct) positive integers and G = Z s p Z s p k F with F free abelian. k If G is a finitely generated abelian group, then the uniquely determined integers m 1,..., m t as in previous theorem, are called the invariant factors of G. The uniquely determined prime powers are called the elementary divisors of G. Corollary Two finitely generated abelian groups G and H are isomorphic if and only if G/G t and H/H t have the same rank and G and H have the same invariant factors (resp. elementary divisors). 2.3 THE KRULL-SCHMIDT THEOREM Every finitely generated abelian group is the direct sum of a finite number of indecomposable groups and these indecomposable summands are uniquely determined up to isomorphism. Definition A group G is indecomposable if G e and G is not the internal direct product of two of its proper subgroups. Thus G is indecomposable if and only if G e and G = H K implies H = e or K = e. For example, every simple group is indecomposable. Also Z, Z p n, and S n are indecomposable but not simple. Definition A group G is said to satisfy the ascending chain condition (ACC) on (normal) subgroups if for every chain G 1 < G 2 < of (normal) subgroups of G there is an integer n such that G i = G n for all i n. G is said to satisfy the descending chain condition (DCC) on (normal) subgroups if for every chain G 1 > G 2 > of (normal) subgroups of G there is an integer n such that G i = G n for all i n. Every finite group satisfies both chain conditions Theorem If a group G satisfies either the ascending or descending chain condition on normal subgroups, then G is the direct product of a finite number of indecomposable subgroups. An endomorphism f of a group G is called a normal endomorphism if af(b)a 1 = f(aba 1 ) for all a, b G. Lemma Let G be a group that satisfies the ascending (resp. descending) chain condition on normal subgroups and f a (normal) endomorphism of G. Then f is an automorphism if and only if f is an epimorphism (resp. monomorphism). 13

20 CHAPTER 2. THE STRUCTURE OF GROUPS Lemma (Fitting) If G is a group that satisfies both the ascending and descending chain conditions on normal subgroups and f is a normal endomorphism of G, then for some n 1, G = Kerf n Imf n. An endomorphism f of a group G is said to be nilpotent if there exists a positive integer n such that f n (g) = e for all g G. Corollary If G is an indecomposable group that satisfies both the ascending and descending chain conditions on normal subgroups and f is a normal endomorphism of G, then either f is nilpotent or f is an automorphism. Corollary Let G ( e ) be an indecomposable group that satisfies both the ascending and descending chain conditions on normal subgroups. If f 1,..., f n are normal nilpotent endomorphisms of G such that every f i1 + + f ir (1 i 1 < i 2 < < i r n) is an endomorphism, then f 1 +f 2 + +f n is nilpotent. Theorem (Krull-Schmidt) Let G be a group that satisfies both (ACC) and (DCC) on normal subgroups. If G = G 1 G 2 G s and G = H 1 H 2 H t with each G i, H j indecomposable, then s = t and after reindexing G i = Hi for every i and for each r < t. G = G 1 G r H r+1 H t. 2.4 THE ACTION OF A GROUP ON A SET Definition An action of a group G on a set S is a function G S S (usually denoted by (g, x) gx) such that for all x S and g 1, g 2 G: ex = x and (g 1 g 2 )x = g 1 (g 2 x) When such an action is given, we say that G acts on the set S. The notation gx is ambiguous. As an example, let H be a subgroup of a group G. An action of H on the set G is given by (h, x) hxh 1 ; to avoid confusion with the product in G, this action of h H is always denoted hxh 1 and not hx. This action of h H on G is called conjugation by h and the element hxh 1 is said to be a conjugate of x. Theorem Let G be a group that acts on a set S. (i) The relation on S defined by x x gx = x for some g G is an equivalence relation. (ii) For each x S, G x = {g G : gx = x} is a subgroup of G. The equivalence classes are called the orbits of G on S; the orbit of x S is denoted x. The subgroup G x is called variously the subgroup fixing x, the isotropy group of x or the stabilizer of x. For some examples, if a group G acts on itself by conjugation, then the orbit {gxg 1 : g G} of x G is called the conjugacy class of x. If a subgroup 14

21 CHAPTER 2. THE STRUCTURE OF GROUPS H acts on G by conjugation the stabilizer group H x = {h H : hxh 1 = x} = {h H : hx = xh} is called the centralizer of x in H and is denoted C H (x). If H = G, C G (x) is simply called the centralizer of x. If H acts by conjugation on the set S of all subgroups of G, then the subgroup of H fixing K S namely {h H : hkh 1 = K} is called the normalizer of K in H and denoted N H (K). The group N G (K) is simply called the normalizer of K. Clearly every subgroup K is normal in N G (K); K is normal in G if and only if N G (K) = G. Theorem If a group G acts on a set S, then the cardinal number of the orbit of x S is the index [G : G x ]. Corollary Let G be a finite group and K a subgroup of G. (i) The number of elements in the conjugacy class x G is [G : C G (x)], which divides G ; (ii) if x 1,..., x n (x i G) are the distinct conjugacy cliasses of G, then. n G = [G : C G (x i )]; i=1 This equation is know as the class equation of the finite group G. (iii) the number of subgroups of G conjugate to K is [G : N G (K)], which divides G. Theorem If a group G acts on a set S, then this action induces a homomorphism G A(S), where A(S) is the group of all permutations of S. Corollary (Cayley) If G is a group, then there is a monomorphism G A(G). Hence every group is isomorphic to a group of permutations. In particular every finite group is isomorphic to a subgroup of S n with n = G. Corollary Let G be a group. (i) For each g G, conjugation by g induces an automorphism of G. (ii) There is a homomorphism G AutG whose kernel is Z(G) = {g G : gx = xg for all x G}. If G acts on itself by conjugation then for each g G, the map φ g : G G given by φ g (x) = gxg 1 is an automorphism, called the inner automorphism induced by g. The normal subgroup Z(G) = Kerφ is called the center of G. Proposition Let H be a subgroup of a group G and let G act on the set S of all left cosets of H in G by left translation. Then the kernel of the induced homomorphism G A(S) is contained in H. Corollary If H is a subgroup of index n in a group G and no nontrivial normal subgroup of G is contained in H, then G is isomorphic to a subgroup of S n. 15

22 CHAPTER 2. THE STRUCTURE OF GROUPS Corollary If H is a subgroup of a finite group G of index p, where p is the smallest prime dividing the order of G, then H is normal in G. 2.5 THE SYLOW THEOREMS Lemma If a group H of order p n (p prime) acts on a finite set S and if S 0 = {x S : hx = x for all h H}, then S S 0 (mod p). Theorem (Cauchy) If G is a finite group whose order is divisible by a prime p, then G contains an element of order p. A group in which every element has order a power ( 0) of some fixed prime p is called a p-group. If H is a subgroup of a group G and H is a p-group, H is said to be a p-subgroup of G. Corollary A finite group G is a p-group if and only if G is a power of p. Corollary The center Z(G) of a nontrivial finite p-group G contains more than one element. Lemma If H is a p-subgroup of a finite group G, then [N G (H) : H] [G : H](mod p). Corollary If H is a p-subgroup of a finite group G such that p divides [G : H], then N G (H) H. Theorem (First Sylow Theorem) Let G be a group of order p n m, with n 1, p prime, and (p, m) = 1. Then G contains a subgroup of order p i for each 1 i n and every subgroup of G of order p i (i < n) is normal in some subgroup of order p i+1. A subgroup P of a group G is said to be a Sylow p-subgroup (p prime) if P is a maximal p-subgroup of G (that is, P < H < G with H a p-group implies P = H). Sylow p-subgroups always exist, though they may be trivial, and every p-subgroup is contained in a Sylow p-subgroup. A finite group G has a nontrivial Sylow p-subgroup for every prime p that divides G. Corollary Let G be a group of order p n m with p prime, n 1 and (m, p) = 1. Let H be a p-subgroup of G. (i) H is a Sylow p-subgroup of G if and only if H = p n. (ii) Every conjugate of a Sylow p-subgroup is a Sylow p-subgroup. (iii) If there is only one Sylow p-subgroup P, then P is normal in G. Theorem (Second Sylow Theorem) If H is a p-subgroup of a finite group G, and P is any Sylow p-subgroup of G, then there exists x G such that H < xp x 1. In particular, any two Sylow p-subgroups of G are conjugate. 16

23 CHAPTER 2. THE STRUCTURE OF GROUPS Theorem (Third Sylow Theorem) If G is a finite group and p a prime, then the number of Sylow p-subgroups of G divides G and is of the form kp + 1 for some k 0. Theorem If P is a Sylow p-subgroup of a finite group G, then N G (N G (P )) = N G (P ). 2.6 CLASSIFICATION OF FINITE GROUPS Proposition Let p and q be primes such that p > q. If q p 1, then every group of order pq is isomorphic to the cyclic group Z pq. If q p 1, then there are (up to isomorphism) exactly two distinct groups of order pq: the cyclic group Z pq and a nonabelian group K generated by elements c and d such that c = p; d = q; dc = c s d, where s 1 (mod p) and s q 1 (mod p). Corollary If p is an odd prime, then every group of order 2p is isomorphic either to the cyclic group Z 2p or the dihedral group D p. 2.7 NILPOTENT AND SOLVABLE GROUPS Definition (i) For any (finite or infinite) group G define the following subgroups inductively: Z 0 (G) = 1, Z 1 (G) = Z(G) and Z i+1 (G) is the subgroup of G containing Z i (G) such that Z i+1 (G) / Z i (G) = Z(G/Z i (G)) (i.e., Z i+1 (G) is the complete preimage in G of the center of G/Z i (G) under the natural projection). The chain of subgroups Z 0 (G) < Z 1 (G) < Z 2 (G) < is called the upper central series of G. (ii) A group G is called nilpotent if Z c (G) = G for some c Z. The smallest such c is called the nilpotence class of G. Proposition Let p be a prime and let P be a group of order p a. Then P is nilpotent of nilpotence class at most a 1. Theorem Let G be a finite group, let p 1, p 2,..., p s be the distinct primes dividing its order and let P i Syl pi (G), 1 i s. Then the following are equivalent: (i) G is nilpotent. (ii) If H < G then H < N G (H), i.e., every proper subgroup of G is a proper subgroup of its normalizer in G. (iii) P i G for 1 i s, i.e., every Sylow subgroup is normal in G. (iv) G = P 1 P 2 P s. 17

24 CHAPTER 2. THE STRUCTURE OF GROUPS Corollary A finite abelian group is the direct product of its Sylow subgroups. Proposition If G is a finite group such that for all positive integers n dividing its order, G contains at most n elements x satisfying x n = 1, then G is cyclic. Proposition (Frattini s Argument) Let G be a finite group, let H be a normal subgroup of G and let P be a Sylow p-subgroup of H. Then G = HN G (P ) and [G : H] divides [N G (P )]. Proposition A finite group is nilpotent if and only if every maximal subgroup is normal. Corollary If G is a finite nilpotent group and m divides G, then G has a subgroup of order m. Theorem The direct product of a finite number of nilpotent groups is nilpotent. Definition Let G be a group. The subgroup of G generated by the set {aba 1 b 1 : a, b G} is called the commutator subgroup of G and denoted G. Theorem If G is a group, then G is a normal subgroup of G and G/G is abelian. If N is a normal subgroup of G, then G/N is abelian if and only if N contains G. Definition For any (finite or infinite) group G define the following subgroups inductively: G (0) = G, G (1) = G and G (i) = (G (i 1) ). G (i) is called the ith derived subgroup of G. The chain of groups G (0) G (1) G (2) is called the lower central series of G. (The term lower indicates that G (i) G (i+1). Each G (i) is a normal subgroup of G. Definition A group G is said to be solvable if G (n) = e for some n. Proposition Every nilpotent group is solvable. Theorem (i) Every subgroup and every homomorphic image of a solvable group is solvable. (ii) If N is a normal subgroup of a group G such that N and G/N are solvable, then G is solvable. Corollary If n 5, then the symmetric group S n is not solvable. Proposition (P. Hall) Let G be a finite solvable group of order mn, with (m, n) = 1. Then (i) G contains a subgroup of order m; (ii) any two subgroups of G of order m are conjugate; (iii) any subgroup of G of order k, where k m, is contained in a subgroup of order m. 18

25 CHAPTER 2. THE STRUCTURE OF GROUPS 2.8 NORMAL AND SUBNORMAL SERIES Definition A subnormal series of a group G is a chain of subgroups G = G 0 > G 1 > > G n such that G i+1 is normal in G i for 0 i < n. The factors of the series are the quotient groups G i /G i+1. The length of the series is the number of strict inclusions (or alternatively, the number of nonidentity factors). A subnormal series such that G i is normal in G for all i, is said to be a normal series. Definition Let G = G 0 > G 1 > > G n be a subnormal series. A one-step refinement of this series is any series of the form G = G 0 > > G i > N > G i+1 > > G n or G = G 0 > > G n > N, where N is a normal subgroup of G i and (if i < n) G i+1 is normal in N. A refinement of a subnormal series S is any subnormal series obtained from S by a finite sequence of one-step refinements. A refinement of S is said to be proper if its length is larger than the length of S. Definition A subnormal series G = G 0 > G 1 > > G n = e is a composition series if each factor G i /G i+1 is simple. A subnormal series G = G 0 > G 1 > > G n = e is a solvable series if each factor is abelian. Theorem (i) Every finite group G has a composition series. (ii) Every refinement of a solvable series is a solvable series. (iii) A subnormal series is a composition series if and only if it has no proper refinements. Theorem A group G is solvable if and only if it has a solvable series. Proposition A finite group G is solvable if and only if G has a composition series whose factors are cyclic of prime order. Definition Two subnormal series S and T of a group G are equivalent if there is a one-to-one correspondence between the nontrivial factors of S and the nontrivial factors of T such that corresponding factors are isomorphic groups. Lemma If S is a composition series of a group G, then any refinement of S is equivalent to S. Lemma (Zassenhaus) Let A, A, B, B be subgroups of a group G such that A is normal in A and B is normal in B. (i) A (A B ) is a normal subgroup of A (A B); (ii) B (A B) is a normal subgroup of B (A B); (iii) A (A B)/A (A B ) = B (A B)/B (A B). Theorem (Schreier) Any two subnormal (resp. normal) series of a group G have subnormal (resp. normal) refinements that are equivalent. Theorem (Jordan-Holder) Any two composition series of a group G are equivalent. Therefore every group having a composition series determines a unique list of simple groups. 19

26 CHAPTER 2. THE STRUCTURE OF GROUPS 20

27 Chapter 3 Rings 3.1 RINGS AND HOMOMORPHISMS Definition A ring is a nonempty set R together with two binary operations (usually denoted as addition (+) and multiplication) such that: (i) (R, +) is an abelian group; (ii) (ab)c = a(bc) for all a, b, c R (associative multiplication); (iii) a(b + c) = ab + ac and (a + b)c = ac + bc (left and right distributive laws). If in addition: (iv) ab = ba for all a, b R then R is said to be a commutative ring. If R contains an element 1 R such that (v) 1 R a = a1 R = a for all a R, then R is said to be a ring with identity. 0. The additive identity element of a ring is called the zero element and denoted Theorem Let R be a ring. Then (i) 0a = a0 = 0 for all a R; (ii) ( a)b = a( b) = (ab) for all a, b R; (iii) ( a)( b) = ab for all a, b R; (iv) (na)b = a(nb) = n(ab) for all n Z and all a, b R; (v) ( n i=1 a i)( m j=1 b j) = n m i=1 j=1 a ib j for all a i, b j R. Definition A nonzero element a in a ring R is said to be a left (resp. right) zero divisor if there exists a nonzero b R such that ab = 0 (resp. ba = 0). A zero divisor is an element of R which is both a left and a right zero divisor. A ring R has no zero divisors if and only if the right and left cancellation laws hold in R; that is, for all a, b, c R with a 0, ab = ac or ba = ca b = c. Definition An element a in a ring R with identity is said to be left (resp. right) invertible if there exists c R (resp. b R) such that ca = 1 R (resp.

28 CHAPTER 3. RINGS ab = 1 R ). The element c (resp. b) is called a left (resp. right) inverse of a. An element a R that is both left and right invertible is said to be invertible or to be a unit. (i) The left and right inverses of a unit a in a ring R with identity necessarily coincide. (ii) The set of units in a ring R with identity forms a group under multiplication. Definition A commutative ring R with identity 1 R 0 and no zero divisors is called an integral domain. A ring D with identity 1 D 0 in which every nonzero element is a unit is called a division ring. A field is a commutative division ring. (i) Every integral domain and every division ring has at least two elements (namely 0 and 1 R ). (ii) A ring R with identity is a division ring if and only if the nonzero elements of R form a group under multiplication. (iii) Every field F is an integral domain. Theorem (Binomial Theorem) Let R be a ring with identity, n a positive integer, and a, b, a 1, a 2,..., a s R. (i) If ab = ba, then (a + b) n = n ( n k=0 k) a k b n k ; (ii) If a i a j = a j a i for all i and j, then (a 1 + a a s ) n = n! (i 1!) (i n!) ai 1 1 a i 2 2 a is s where the sum is over all s-tuples (i 1, i 2,..., i s ) such that i 1 + i i s = n. Definition Let R and S be rings. A function f : R S is a homomorphism of rings provided that for all a, b R:. f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b) A homomorphism of rings is, in particular, a homomorphism of the underlying additive groups. The same terminology is used: monomorphism, epimorphism, isomorphism, etc. A monomorphism of rings R S is sometimes called an embedding of R in S. The kernel of a homomorphism of rings f : R S is its kernel as a map of additive groups; that is ker f = {r R : f(r) = 0}. Similarly the image of f, denoted Im f, is {s S : s = f(r) for some r R}. If R and S both have identities 1 R and 1 S, we do not require that a homomorphism of rings map 1 R to 1 S. Definition Let R be a ring. If there is a least positive integer n such that na = 0 for all a R, then R is said to have characteristic n. If no such n exists R is said to have characteristic zero. (Notation: char R = n). 22

29 CHAPTER 3. RINGS Theorem Let R be a ring with identity 1 R and characteristic n > 0. (i) If φ : Z R is the map given by m m1 R, then φ is a homomorphism of rings with kernel n = {kn : k Z}. (ii) n is the least positive integer such that n1 R = 0. (iii) If R has no zero divisors (in particular if R is an integral domain), then n is prime. Theorem Every ring R may be embedded in a ring S with identity. The ring S (which is not unique) may be chosen to be either of characteristic zero or of the same characteristic as R. 3.2 IDEALS Definition Let R be a ring and S a nonempty subset of R that is closed under the operations of addition and multiplication in R. If S is itself a ring under these operations then S is called a subring of R. A subring I of a ring R is a left ideal provided I is a right ideal provided r R and x I rx I; r R and x I xr I; I is an ideal if it is both a left and right ideal. Two ideals of a ring R are R itself and the trivial ideal (denoted 0). A (left) ideal I of R such that I 0 and I R is called a proper (left) ideal. Observe that if R has an identity 1 R and I is a (left) ideal of R, then I = R if and only if 1 R I. Consequently, a nonzero (left) ideal I of R is proper if and only if I contains no units of R; (for if u R is a unit and u I, then 1 R = u 1 u I). In particular, a division ring D has no proper left (or right) ideals since every nonzero element of D is a unit. Theorem A nonempty subset I of a ring R is a left (resp. right) ideal if and only if for all a, b I and r R: (i) a, b I a b I; and (ii) a I, r R ra I (resp. ar I). Corollary Let {A i : i I} be a family of (left) ideals in a ring R. Then i I A i is also a (left) ideal Definition Let X be a subset of a ring R. Let {A i : i I} be the family of all (left) ideals in R which contain X. Then i I A i is called the (left) ideal generated by X. This is denoted (X). 23

30 CHAPTER 3. RINGS The elements of X are called generators of the ideal (X). If X = {x 1,..., x n }, then the ideal (X) is denoted (x 1, x 2,..., x n ) and said to be finitely generated. An ideal (x) generated by a single element is called a principal ideal. A principle ideal ring is a ring in which every ideal is principle. A principle ideal ring which is an integral domain is called a principle ideal domain. Theorem Let R be a ring a R and X R. (i) The principle ideal (a) consists of all elements of the form ra + as + na + m i=1 r ias i (r, s, r i, s i R; m N ; and n Z). (ii) If R has an identity, then (a) = { n i=1 r ias i : r i, s i R; n N }. (iii) If a is in the center of R, then (a) = {ra + na : r R, n Z}. (iv) Ra = {ra : r R} (resp. ar = {ar : r R}) is a left (resp. right) ideal in R (which may not contain a). If R has an identity, then a Ra and a ar. (v) If R has an identity and a is in the center of R, then Ra = (a) = ar. (vi) If R has an identity and X is in the center of R, then the ideal (X) consists of all finite sums r 1 a r n a n (n N ; r i R; a i X). Let A 1, A 2,..., A n be nonempty subsets of a ring R. Denote by A 1 + A A n the set {a 1 + a a n : a i A i for i = 1, 2,..., n}. If A and B are nonempty subsets of R let AB denote the set of all finite sums {a 1 b 1 + +a n b n : n N ; a i A; b i B}. More generally let A 1 A 2, A n denote the set of all finite sums of elements of the form a 1 a 2... a n (a i A i for i = 1, 2,..., n). In the special case when all A i (1 i n) are the same set A we denote A 1 A 2 A n = AA A by A n. Theorem Let A, A 1, A 2,..., A n, B and C be (left) ideals in a ring R. (i) A 1 + A A n and A 1 A 2 A n are (left) ideals; (ii) (A + B) + C = A + (B + C); (iii) (AB)C = ABC = A(BC); (iv) B(A 1 + A A n ) = BA 1 + BA 2 + BA n ; and (A 1 + A A n )C = A 1 C + A 2 C + + A n C. Ideals play approximately the same role in the theory of rings as normal subgroups do in the theory of groups. For instance, let R be a ring and I an ideal of R. Since the additive group of R is abelian, I is a normal subgroup. Consequently, there is a well-defined quotient group R/I in which addition is given by (a + I) + (b + I) = (a + b) + I R/I can in fact be made into a ring. Theorem Let R be a ring and I an ideal of R. Then the additive quotient group R/I is a ring with multiplication given by 24

31 CHAPTER 3. RINGS (a + I)(b + I) = ab + I. If R is commutative or has an identity, then the same is true of R/I. Theorem If f : R S is a homomorphism of rings, then the kernel of f is an ideal in R. Conversely, if I is an ideal in R, then the map π : R R/I given by r r +I is an epimorphism of rings with kernel I. The map π is called the canonical epimorphism. Theorem If f : R S is a homomorphism of rings and I is an ideal of R which is contained in the kernel of f, then there is a unique homomorphism of rings f : R/I S such that f(a + I) = f(a) for all a R. Imf = Imf and ker f = (ker f)/i. f is an isomorphism if and only if f is an epimorphism and I = ker f. Corollary (First Isomorphism Theorem) If f : R S is a homomorphism of rings, then f induces an isomorphism of rings R/ ker f = Imf. Corollary If f : R S is a homomorphism of rings, I is an ideal in R and J is an ideal in S such that f(i) J, then f induces a homomorphism of rings f : R/I S/J, given by a + I f(a) + J. f is an isomorphism if and only if Imf + J = S and f 1 (J) I. In particular, if f is an epimorphism such that f(i) = J and ker f I, then f is an isomorphism. Theorem Let I and J be ideals in a ring R. (i) (Second Isomorphism Theorem) There is an isomorphism of rings I/(I J) = (I + J)/J; (ii) (Third Isomorphism Theorem) if I J, then J/I is an ideal in R/I and there is an isomorphism of rings (R/I)/(J/I) = R/J. Theorem If I is an ideal in a ring R, then there is a one-to-one correspondence between the set of all ideals of R which contain I and the set of all ideals of R/I, given by J J/I. Hence every ideal in R/I is of the form J/I, where J is an ideal of R which contains I. Definition An ideal P in a ring R is said to be prime if P R and for any ideals A, B in R AB P A P or B P. Theorem If P is an ideal in a ring R such that P R and for all a, b R ab P a P or b P, then P is prime. Conversely if P is prime and R is commutative, then P satisfies previous condition. 25

32 CHAPTER 3. RINGS Theorem In a commutative ring R with identity 1 R 0 an ideal P is prime if and only if the quotient ring R/P is an integral domain. Definition An ideal (resp. left ideal) M in a ring R is said to be maximal if M R and for every ideal (resp. left ideal) N such that M N R, either N = M or N = R. Theorem In a nonzero ring R with identity, maximal (left) ideals always exist. In fact, every (left) ideal in R (except R itself) is contained in a maximal (left) ideal. Theorem If R is a commutative ring with identity, then every maximal ideal M in R is prime. Theorem Let M be an ideal in a ring R with identity 1 R 0. (i) If M is maximal and R is commutative, then the quotient ring R/M is a field. (ii) If the quotient ring R/M is a division ring, then M is maximal. Corollary The following conditions on a commutative ring R with identity 1 R 0 are equivalent. (i) R is a field; (ii) R has no proper ideals; (iii) 0 is a maximal ideal in R; (iv) every nonzero homomorphism of rings R S is a monomorphism. Theorem Let {R i : i I} be a nonempty family of rings and i I R i the direct product of the additive abelian groups R i ; (i) i I R i is a ring with multiplication defined by {a i } i I {b i } i I = {a i b i } i I ; (ii) if R i has an identity (resp. is comm) for every i I, then i I R i has an identity (resp. is comm); (iii) for each k I the canonical projection π k : i I R i R k given by {a i } a k, is an epimorphism of rings; (iv) for each k I the canonical injection i k : R k i I R i, given by a k {a i } (where a i = 0 for i k), is a monomorphism of rings. Theorem Let {R i : i I} be a nonempty family of rings, S a ring and {φ i : S R i : i I} a family of homomorphisms of rings. Then there is a unique homomorphism of rings φ : S i I R i such that π i φ = φ i for all i I. The ring i I R i is uniquely determined up to isomorphism by this property. Theorem Let A 1, A 2,..., A n be ideals in a ring R such that (i) A 1 + A A n = R and (ii) for each k(1 k n), A k (A A k 1 + A k+1 + +A n ) = 0. Then there is a ring isomorphism R = A 1 A 2 A n. If R is a ring and A 1,..., A n are ideals in R that satisfy previous theorem, then R is said to be the internal direct product of the ideals A i. Similar to groups, 26