# 1 Finite abelian groups

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1 Last revised: May 16, 2014 A.Miller M542 miller/ Each Problem is due one week from the date it is assigned. Do not hand them in early. Please put them on the desk in front of the room at the beginning or end of class. Include the statement of the problem as part of your solution. The date the problem was assigned in class is in parentheses. 1 Finite abelian groups Theorem 1.1 (Chinese remainder theorem) Given n, m relatively prime integers for every i, j Z there is an x Z such that x = i mod n and x = j mod m. Problem 1 (Fri Jan 24) (a) Find an integer x such that x = 6 mod 10 and x = 15 mod 21 and 0 x 210. (b) Find the smallest positive integer y such that y = 6 mod 10 and y = 15 mod 21 and y = 8 mod 11. Problem 2 (Fri Jan 24) (a) Find integers i, j such that there is no integer x with x = i mod 6 and x = j mod 15. (b) Find all pairs i, j with i = 0, 1,... 5 and j = 0, 1,..., 14 such that there is an integer x with x = i mod 6 and x = j mod 15. Theorem 1.2 Z n Z m Z nm iff n, m are relatively prime. Lemma 1.3 Suppose n, m are relatively prime, G is a finite abelian group such that x nm = e for every x G. Let G n = {x G : x n = e} and G m = {x G : x m = e}. Then G n and G m are subgroups of G, G n G m = {e}, G n G m = G, and therefore 1

2 G G n G m Corollary 1.4 (Decomposition into p-groups) Suppose G is an abelian group and G = p i 1 1 p i 2 2 p in n where p 1 < p 2 < < p n are primes. Then G G 1 G 2 G n where for each j if x G j then x n j = e where n j = p i j j. Problem 3 (Mon Jan 27) Prove that for any n there is only one abelian group (up to isomorphism) of size n iff n is square-free. Square-free mean that no p 2 divides n for p a prime. Lemma 1.5 Suppose G is a finite abelian p-group and a G has maximum order, then there exists a subgroup K G such that a K = G and a K = {e}. The proof given in class is like the one in Gallian or Judson. Theorem 1.6 Any finite abelian group is isomorphic to a product of cyclic groups each of which has prime-power order. Problem 4 (Wed Jan 29) Let G be a finite abelian group. Prove that the following are equivalent 1. For every subgroup H of G there is a subgroup K of G with HK = G and H K = {e}. 2. Every element of G has square-free order. Hint: Polya s Dictum: If you can t do a problem, then there is an easier problem you can t do. Find it. Lets call property (1) the Complementation Property for G or CP for short. Here are some easier problems: (a) Prove that C p 2 fails to have CP. (b) Prove that C p C p has CP. 2

3 (c) Let G and H be relatively prime. Prove that G H has CP iff both G and H have CP. Theorem 1.7 (Uniqueness) Suppose C p n 1 C p n 1 C p n k C p m 1 C p m 1 C p m l where n 1 n 2 n k 1 and m 1 m 2 m l 1. Then k = l and n i = m i for all i. Problem 5 (Fri Jan 31) How many abelian groups of order 144 are there up to isomorphism? Explain. Problem 6 (Mon Feb 3) Suppose G 1, G 2, H 1, H 2 are finite abelian groups, G 1 G 2 H 1 H 2 and G 1 H 1. Prove that G 2 H 2. Give a counterexample if the word finite is dropped, i.e., G 1 G 2 H 1 H 2 and G 1 H 1 but G 2 is not isomorphic to H 2. I do not know if problem 6 is true or false for finite non-abelian groups. 2 Group Actions and Sylow Theorems For the group G acting on the set X the orbit of a X is orb(a) = def {ga : g G} X. Proposition 2.1 Orbits are either disjoint or the same. Problem 7 (Wed Feb 5) Prove or disprove: For any finite abelian groups G 1 and G 2 with subgroups, H 1 G 1 and H 2 G 2 such that H 1 H 2, if G 1 /H 1 G 2 /H 2 then G 1 G 2. For a given group action of group G on set X, define Stab(a) = {g G : ga = a} for each a X. Called stabilizer or fixed subgroup. Proposition 2.2 Stab(a) is a subgroup of G. Problem 8 (Wed Feb 5) Prove that Stab(ga) = g Stab(a) g 1. 3

4 For H G a subgroup the index of H, [G : H] is the number of H-cosets, {gh : g G}. Lagrange s Theorem says G = [G : H] H. Proposition 2.3 (Orbit-stabilizer formula) orb(a) = [G : Stab(a)]. The conjugacy action of G on G is given by (g, h) ghg 1. Under this action the orbits are called the conjugacy classes. Z(G) the center of G is the subgroup of all elements of G which commute with every other element of g. Equivalently it is the set of elements of G with orbits (conjugacy classes) of size one. C(g) = Stab(g) is called the centralizer subgroup of g. Theorem 2.4 (Class formula) If conj(g 1 ),, conj(g n ) are the conjugacy classes of size greater than one, then G = Z(G) + n [G : C(g k )] Theorem 2.5 (Cauchy) If p is a prime which divides G, then G has an element of order p. Problem 9 This is due in lecture on valentines day. It will be graded in class so do not hand-in. (a) Suppose G is a finite abelian group which contains an element which has non-square-free order. Prove that for some prime p it has an element of order p 2. (b) Suppose a is an element of a finite abelian group G with order p 2 let b = a p, let H =< b > be the subgroup generated by b and suppose K is a subgroup of G with K H = {e}. Prove that a is not an element of HK. (c) Suppose G 1, G 2 are finite abelian groups with G 1 and G 2 relatively prime. Show that for any subgroup H G 1 G 2 there are subgroups H 1 G 1 and H 2 G 2 such that H = H 1 H 2. (Warning: the relatively prime hypothesis is necessary.) (d) Suppose G 1, G 2 are finite abelian groups with G 1 and G 2 relatively prime. Show that if G 1 and G 2 both have the CP then G 1 G 2 has CP. 1 (e) Prove that C p C p C p has the CP. (f) Prove Problem 4. 1 CP is defined after Problem 4. k=1 4

5 Corollary 2.6 Groups of order p 2 are abelian. Theorem 2.7 (Sylow 1) If G is a finite group and p n divides G, then there exists a subgroup H G with H = p n. Proposition 2.8 Any two n-cycles in S N are conjugates. If τ = c 1 c 2 c n and ρ = c 1c 2 c n are disjoint cycle decomposition with c i = c i all i, then τ and ρ are conjugates. Similarly for the converse. Problem 10 (Mon Feb 10) Prove for any n 3 that Z(S n ) = {id}. Definition 2.9 H G is a p-subgroup iff its order is a power of p. P G is a p-sylow subgroup of G iff P = p n where G = p n m and p does not divide m. Lemma 2.10 Suppose P is a p-sylow subgroup of G, g G has order a power of p, and gp g 1 = P. Then g P. Theorem 2.11 (Sylow 2) If G is a finite group, H a p-subgroup of G, and P a p-sylow subgroup of G, then there exists g G such that H gp g 1. Corollary 2.12 Let G be a finite group such that p divides G. (a) Any p-subgroup of G is contained in a p-sylow subgroup of G. (b) Any two p-sylow subgroups of G are conjugates. (c) Any two p-sylow subgroups of G are isomorphic. (d) A p-sylow subgroup of G is normal iff it is the only p-sylow subgroup of G. Theorem 2.13 (Sylow 3) If G = p n m where p does not divide m and n(p) is the number of p-sylow subgroups of G, then: (a) n(p) = [G : N(P )] for any P a p-sylow subgroup of G, (b) n(p) divides m, and (c) n(p) = 1 mod p Problem 11 (Wed Feb 12) (a) Prove that there are no simple groups of order either 575 or 272. (b) For any prime p prove there are no simple groups of order p(p 1) or p(p + 2). 5

6 Theorem 2.14 If p < q are primes and q is not 1 mod p, then every group of order pq is abelian. Problem 12 (Fri Feb 14) Question (August J.) Suppose every subgroup of finite group G is a normal subgroup. Must G be abelian? Problem 13 (Fri Feb 14) (a) Suppose P is a p-sylow subgroup of G and H a subgroup such that P H and H G. Prove that P G. (b) If K H and H G, does it follow that K G? Show that the answer is No. Consider G = S 4, K = {id, σ} where σ = (12)(34) and H = {id, σ, τ, ρ} where τ and ρ are conjugates of σ. Determine what τ and ρ are and show that K H and H G, but K is not a normal subgroup of G. Theorem 2.15 aut(z p, + p ) is isomorphic to (Z p \{0}, p ) the multiplicative group of nonzero elements. Example 2.16 If p < q are primes and q = 1 mod p, then there is a twisted product of Z p and Z q which has order pq and is not abelian. Problem 14 (Mon Feb 17) Suppose for every x G that x 2 = e. Prove that G is abelian. Problem 15 (Mon Feb 17) Suppose H G is subgroup of index 2, i.e., [G : H] = 2. Prove that it is a normal subgroup of G. Theorem 2.17 If p < q are primes and q = 1 mod p, then up to isomorphism there is a unique nonabelian group of order pq. 3 Polynomials and finite field extensions Theorem 3.1 Suppose that p(x) is a polynomial over the field F and for some α F p(α) = 0. Then p(x) = (x α)q(x) for some polynomial q(x). Corollary 3.2 Any polynomial p F [x] of degree n with more than n roots must be identically zero. Theorem 3.3 Let the exponent of G be the least n such that x n = e for every x G. If G is finite abelian group then G is cyclic iff exp(g) = G. 6

7 Corollary 3.4 The multiplicative group of a finite field is cyclic. Problem 16 (Wed Feb 19) For F a finite field call a F a generator of F iff every nonzero element of F is a power of a. (a) Find a generator of Z 7. (b) How many generators does Z 17 have? (c) How many generators does Z 31 have? 4 Vector spaces over an abstract field Before taking up finite field extensions we review some elementary results on vector spaces. See: Lemma 4.1 (Exchange Lemma) Suppose span(a B) = V and a is not in span(a). Then there exists b B such that span(a {a} (B \ {b})) = V. Theorem 4.2 Every vector space has a basis. Any two bases have the same cardinality. Any set of n + 1 vectors in a vector space of dimension n is linearly dependent. Corollary 4.3 Any finite field F of characteristic p has cardinality p n for some integer n. Problem 17 (Fri Feb 21) Prove that v 1, v 2,..., v n are linearly dependent iff v 1 = 0 or v i+1 span{v 1, v 2,..., v i } for some i with 1 i < n. 5 Extension fields Theorem 5.1 (Kronecker) If p(x) F [x] is a non-constant polynomial, then there exists a field E F and α E with p(α) = 0. Corollary 5.2 (Kronecker) If p(x) F [x] is a polynomial of degree n, then there exists a field E F and α i E such that p(x) = a(x α 1 )(x α 2 ) (x α n ) 7

8 Theorem 5.3 If p(x) F [x] is irreducible and α, β are roots in some extension fields of F then F (α) and F (β) are isomorphic via an isomorphism which fixes F. Corollary 5.4 If p(x) F [x] is irreducible and splits in an extension field E of F then the multiplicity of each root of p is the same. Problem 18 (Mon Feb 24) Let R be a commutative ring with 1. Let I be a maximal ideal in R. Suppose ab = 0. Prove that a I or b I. Problem 19 (Mon Feb 24) Consider p(x) = x 3 + x + 1 as a polynomial in Z 2 [x]. Suppose p has a root α is in some field extension. Construct the multiplication table for Z 2 [α] = def {a + bα + cα 2 : a, b, c Z 2 } Problem 20 (Wed Feb 26) Let α be transcendental over Z 2. Let F = Z 2 (α) and let p(x) = x 2 α. (a) Prove that p is irreducible over F. (b) Prove that if β is a root of p in some some extension field, then p(x) = (x β) 2. (c) Suppose that F is a finite field of characteristic 2. Prove that for every a F there is a b F such that b 2 = a. (d) Suppose that F is a finite field of odd characteristic. Prove that there exists a F for every b F such that b 2 a. (e) Find a field F and an irreducible polynomial p(x) of degree three such that in any extension field in which p splits there exist a β such that p(x) = (x β) 3. Theorem 5.5 The formal derivative for an abstract polynomial f(x) F [x] satisfies the usual derivative laws: (a) If a F and f F [x], then (af) = af. (b) If f, g F [x], then (f + g) = f + g. (c) If f, g F [x], then (fg) = f g + fg. Problem 21 (Fri Feb 28) Prove that the formal derivative for polynomials in F [x] satisfies 8

9 (a) The power rule: (f n ) = n(f n 1 )f (b) The chain rule: f(g(x)) = f (g(x))g (x) Theorem 5.6 For any α F and f F [x] α is repeated root of f iff it is a root of f. Corollary 5.7 The roots of an irreducible polynomial in a field of characteristic zero, are always distinct. Lemma 5.8 If E is any field of characteristic p, then for any α, β E (α + β) pn = α pn + β pn Theorem 5.9 For any p n and there is a field F with F = p n. Problem 22 ((Mon ) Mar 3) Prove for any prime p and positive integer n p n that p divides for any k with 0 < k < p k n. Definition 5.10 For fields F E define [E : F ] to be the dimension of E viewed as a vector space over F. Theorem 5.11 For fields F K E [E : F ] = [E : K] [K : F ] Theorem 5.12 For p(x) F [x] irreducible and α a root of p in some extension field, [F [α] : F ] is the degree of p. Theorem 5.13 If E F is the splitting field of some polynomial in F [x], then [E : F ] is finite. Theorem 5.14 If [E : F ] is finite and α E, then there is an irreducible polynomial p F [x] with p(α) = 0. Problem 23 (Wed Mar 5) p is a prime and n a positive integer. Prove: (a) If F is a field such that F = p n and m is a positive integer then there is a field E with F E and E = p nm. (b) If F E are fields, F = p n and E = p N, then n divides N. 9

10 6 Algebraic closure Definition 6.1 α is algebraic over F iff it is the root of a nontrivial polynomial in F [x]. A field K is algebraically closed iff every nonconstant polynomial f K[x] has a root in K. Theorem 6.2 If F E are fields define Then K is a field and F K F. K = {α E : α is algebraic over F } Steinitz proved that every field F is a subfield of an algebraically closed field K. This requires the Axiom of Choice. Theorem 6.3 Suppose F K and K is algebraically closed. Let E be the elements of K which are algebraic over F. Then E is algebraically closed. 7 Compass and straight-edge Theorem 7.1 (Wantzel 1837) Let C R R be the smallest set containing (0, 0) and (1, 0) and closed under constructions using straight edge and compass. Then C = F c F c where F c is the smallest subfield of R which closed under square roots. Lemma 7.2 For any α α F c iff for some n there are fields F k for k = 0, 1,..., n with α F n and such that F 0 = Q and for each k < n F k+1 = F k [ a k ] for some a k F k. Theorem 7.3 For any α F c [Q[α] : Q] = 2 n for some integer n. Corollary / F c so it is impossible to double the cube. Corollary 7.5 cos(20 ) / F c so it is impossible to trisect every angle. Corollary 7.6 Since π is transcendental and every element of F c is algebraic, it is impossible to square the circle. Problem 24 (Wed Mar 26) Prove or disprove: Using a straight edge and compass it is possible to construct an equilateral triangle with area 1. 10

11 8 Irreducibility criterion Lemma 8.1 (Gauss s Lemma) Suppose f Z[x], then f is irreducible in Q[x] iff f is irreducible in Z[x]. Lemma 8.2 (Eisenstein s Criterion) Suppose f Z[x] has degree n f(x) = a 0 + a 1 x + a 2 x a n x n and for some prime p (a) p does not divide a n, (b) p divides a k for all k = 0, 1,..., n 1, and (c) p 2 does not divide a 0. Then f is irreducible in Z[x]. Theorem 8.3 For any prime p the polynomial f(x) = 1+x+x 2 + +x p 1 is irreducible in Q[x]. Proposition 8.4 If 2 m + 1 is prime, then m is a power of 2. Theorem 8.5 (Gauss) If the regular p-gon is constructible with straight edge and compass, then p = 2 2n + 1 for some integer n. Problem 25 (Fri Mar 28) Prove that [F c : Q] is infinite. F c is the field of constructible reals (straight edge and compass). Problem 26 (Fri Mar 28) Prove that if 2 m 1 is prime, then m is prime. 9 Solvability by radicals For Tartaglia method of solving a cubic polynomial see For a brief history see: Problem 27 (Mon Mar 31) Find the roots of x 3 + 3x 2 + 6x + 5 = 0 using addition, subtraction, multiplication, division, and extraction of roots, i.e., solvability by radicals. 11

12 Theorem 9.1 (Steinitz 1910) Suppose F E are fields of characteristic 0 and [E : F ] is finite. Then there exists α E such that E = F [α]. The same is true if E is a finite field. Example 9.2 There is a field F and α, β with [F [α, β] : F ] finite but there is no γ with F [α, β] = F [γ]. See Problem 28 (Wed Apr 2) Suppose [F [α] : F ] = n, [F [β] : F ] = m, and gcd(n, m) = 1. Prove that [F [α, β] : F ] = nm. 10 Galois Theory Proofs and definitions can be found in galois.pdf see: Proposition 10.1 (2.4 galois.pdf) aut(e F ) is a group. Furthermore, if F E K are fields, then aut(k E) is a subgroup of aut(k F ). Lemma 10.2 (2.5 galois.pdf) Suppose σ, ρ aut(f (α) F ). Then σ = ρ iff σ(α) = ρ(α). Similarly, if σ, ρ aut(f (α 1, α 2,..., α n ) F ) then σ = ρ iff σ(α k ) = ρ(α k ) for all k = 1, 2,..., n. Theorem 10.3 (2.6 galois.pdf) Suppose that K is the splitting field of a polynomial in F [x] of degree n. Then aut(k F ) is isomorphic to a subgroup of S n. Problem 29 (Fri Apr 4) Prove the following: (a) Suppose α + β is algebraic over F, then α is algebraic over F [β]. (b) Suppose α + β and αβ are both algebraic over F, then α is algebraic over F. Definition 10.4 For fields F K we say that K is a splitting field over F iff K is the splitting field of some polynomial in F [x]. Every polynomial in Q[x] splits in C but C is not a splitting field over Q. 12

13 Lemma 10.5 (Extension Lemma 2.9 galois.pdf) Suppose that F F 1 K and F F 2 K are fields, K is a splitting field over F, and σ : F 1 F 2 is an isomorphism which fixes F. Then there exists ρ : K K an automorphism which extends σ. Theorem 10.6 (3.1 galois.pdf) Suppose F K, K is a splitting field over F, p F [x] is irreducible, and there is α K such that p(α) = 0. Then p splits in K. Problem 30 (Mon Apr 7) Suppose that F K 1 L and F K 2 L and K 1 and K 2 are splitting fields over F. Prove that K 1 K 2 is a splitting field over F. Theorem 10.7 (2.8 galois.pdf) Suppose F K, K is a splitting field over F, and these fields have characteristic zero. Then aut(k, F ) = [K : F ]. Theorem 10.8 (2.10 galois.pdf) Suppose F K E, K and E are splitting fields over F. Then aut(e K) aut(e F ) and aut(e F ) aut(e K) aut(k F ) Proposition 10.9 Suppose F K E, E is a splitting fields over F, and aut(e K) aut(e F ). Then K is a splitting field over F. Problem 31 (Wed Apr 9) Suppose that E is a splitting field over F and p F [x] splits in E as p(x) = (x α 1 )(x α 2 ) (x α n ) where α i α j whenever i j. Prove that the following are equivalent: (1) p is irreducible. (2) for any i, j there is σ aut(e F ) such σ(α i ) = α j. Theorem (5.3 galois.pdf) Suppose F E is a radical Galois extension, then aut(e F ) is a solvable group. Example If 2 generates the multiplicative group of Z p, then is irreducible over Z 2. f(x) = 1 + x + x x p 1 13

14 See Gurrier See also Artin s conjecture on primitive roots Theorem (5.4 galois.pdf) Subgroups of solvable groups are solvable and homomorphic images of solvable groups are solvable. Problem 32 (Mon Apr 14) For each of the following polynomials compute its Galois group, i.e., aut(k Q) where K is the splitting field of the polynomial. (a) x 5 1 (b) x 4 2 (c) x 4 2x 2 2 Theorem Suppose K is the splitting field of a polynomial in F 1 [x] and F 1 F 2 F m satisfies F 1 K F m and F k+1 is a radical Galois extension of F k for each k < m. Then aut(k F 1 ) is a solvable group. Lemma (a) {(i, i + 1) : 1 i < n} generates S n. (adjacent swaps) (b) {(1, 2), (1, 2, 3,..., n)} generates S n. (c) {(1, i), (1, 2, 3,..., n)} generates S n if n is prime. (d) If n is prime, then any subgroup of S n which contains an n-cycle and at least one transposition must be S n. Problem 33 (Wed Apr 16) In the Lemma (d) must n be prime? Prove or disprove: {(1, 2, 3, 4), (1, 3)} generates S 4. Theorem Suppose f(x) Q[x] is an irreducible polynomial of prime degree p such that f has exactly p 2 real roots. If K is the splitting field of f, then aut(k, Q) is isomorphic to S p. Example If f(x) = x 5 5x three real roots. then f is irreducible and has exactly Theorem The alternating group A 5 is simple. Hence S 5 is not solvable. 14

15 Corollary There is polynomial in Q[x] of degree 5 which cannot be solved by radicals. Problem 34 (Mon Apr 21) σ, τ A 5 are conjugate in A 5 iff there is ρ A 5 such that σ = ρ 1 τρ. Prove that every element of A 5 except the identity is conjugate to exactly one of the following: (a) (1, 2, 3) (b) (1, 2)(3, 4) (c) (1, 2, 3, 4, 5) (d) (2, 1, 3, 4, 5) In particular (c) and (d) are not conjugates. Theorem For any n there are fields E K such that K is the splitting field of a polynomial in E[x] and aut(k E) is isomorphic to S n. Theorem (char 0) Suppose F K and K is the splitting field of a polynomial in F [x] and H aut(k F ) is a subgroup. Then there exists a field E with F E K and aut(k E) = H. Corollary Every finite group is a Galois group. Proof of the fundamental theorem of algebra using Galois theory: Problem 35 (Wed Apr 23) Let K Q be the splitting field of f(x) = (x 2 + 1) (x 2 2) = x 4 x 2 2 (a) Prove that aut(k Q) Z 2 Z 2 (b) Find an irreducible polynomial p(x) whose splitting field is K. Definition A polynomial f(x) Q[x] is solvable by real radicals iff its roots are in the smallest subfield S R which is closed under taking real roots, i.e., if a S, a > 0 and n N then n a S. Lemma Suppose F C is a subfield, p a prime, and a F. Then f(x) = x p a is reducible in F iff it has a root in F. Theorem Suppose f(x) Q[x] is an irreducible cubic with three real roots. Then f(x) is not solvable by real radicals. 15

16 11 Similar Matrices For this material see Problem 36 (Mon Apr 28) Suppose A and B are matrices with real entries and there exists a matrix P with complex entries such that A = P BP 1. Prove there exists a matrix P with real entries such that A = P BP 1. Hint: Show {Q : AQ = QB} is a subspace. Theorem 11.1 Suppose F is an infinite field, A and B are F -matrices, and for some field extension E F there is an E-matrix P such that A = P BP 1. Then there is an F -matrix P such that A = P BP 1. For algebraically closed fields A and B are similar iff they have the same Jordan Normal forms up to a permutation of the Jordan blocks. So without loss we may as well assume that E is the algebraic closure of F. By adding one new element at a time it suffices to prove the Theorem for E = F [α] with [E : F ] finite. Let p(x) F [x] be the minimal polynomial for α. Consider the vector space M = {P : AP = P B} where the P are E-matrix. Note that any such P has entries which are a polnomial in α. So we can write P = P 0 + αp α n P n where the P i are F -matrices. Let f(x) F [x] be the determinate of P 0 + xp x n P n Since f(α) 0 and F is an infinite field there is an β F such that f(β) 0. The F -matrix P = P 0 + βp β n P n is invertible and witnesses the similarity of A and B. 16

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