MTH Abstract Algebra II S17. Review for the Final Exam. Part I
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1 MTH Abstract Algebra II S17 Review for the Final Exam Part I You will be allowed to use the textbook (Hungerford) and a print-out of my online lecture notes during the exam. Nevertheless, I recommend that you memorize the following definitions and theorems.: Lemma Let G be a group and a, b, c G. Then ab = ac b = c ba = ca Subgroup Proposition only if Let G be a group and H a subset of G. Then H is subgroup of G if and (i) H is closed under, that is a b H for all a, b H. (ii) e G H. (iii) H is closed under inverses, that is a 1 H for all a H. Lagrange Theorem In particular, K divides G. Let G be a finite group and K a subgroup of G. Then G = K G/K Definition Let N be a subgroup of the group G. N is called a normal subgroup of G and we write N G provided that for all g G. gn = Ng Lemma Let N be a subgroup of the group G. Then the following statements are equivalent: (a) N is normal in G. (f) N is invariant under conjugation, that is ana 1 N for all a G and n N. (g) Every right coset of N is a left coset of N. Corollary Let N be a normal subgroup of the group G and a, b G. (a) (an)(bn) = abn. (d) (an) 1 = a 1 N. 1
2 First Isomorphism Theorem Let φ G H be a homomorphism of groups. Then φ G/ ker φ Im φ, g ker φ φ(g) is well-defined isomorphism of groups. In particular G/ ker φ Im φ Second Isomorphism Theorem Let G be a group, N a normal subgroup of G and A a subgroup of G. Then A N is a normal subgroup of A, AN is a subgroup of G, N is a normal subgroup of AN and the map A/A N AN/N, a(a N) an is a well-defined isomorphism. In particular, A/A N AN/N. Correspondence Theorem Let N be a normal subgroup of the group G. Put S(G, N) = {H N H G} and S(G/N) = {F F G/N}. Let be the natural homomorphism. (a) Let N K G. Then π(k) = K/N. π G G/N, g gn (b) Let F G/N. Then π 1 (F ) = T F T. (c) Let N K G and g G. Then g K if and only if gn K/N. (d) The map is a well-defined bijection with inverse In other words: β S(G, N) S(G/N), K K/N α S(G/N) S(G, N), F π 1 (F ). (a) If N K G, then K/N is a subgroup of G/N. (b) For each subgroup F of G/N there exists a unique subgroup K of G with N K and F = K/N. Moreover K = π 1 (F ). (e) Let N K G. Then K G if and only if K/N G/N. (f) Let N H G and N K G. Then H K if and only if H/N K/N. (g) (Third Isomorphism Theorem) Let N H G. Then the map is a well-defined isomorphism. ρ G/H (G/N)/(H/N), gh (gn) (H/N) 2
3 Definition Let G be group and I a set. An action of G on I is a function such that (act:i) e i = i for all i I. (act:ii) g (h i) = (g h) i for all g, h G, i I. G I I (g, i) (g i) The pair (I, ) is called a G-set. We also say that G acts on I via. Abusing notations we often just say that I is a G-set. Also we often just write gi for g i. Lemma Let G be a group and I a set. (a) Suppose is an action of G on I. For a G define Then f a Sym(I) and the map f a I I, i a i is a homomorphism. Φ G Sym(I), a f a Isomorphism Theorem for G-sets Let G be a group and (I, ) a G-set. Let i I and put H = Stab G (i). Then φ G/H Gi, ah ai is a well-defined G-isomorphism. In particular G/H G Gi, Gi = G/Stab G (i) and Gi divides G Orbits Equation Let G be a group acting on a finite set I. Let I k, 1 k n be the distinct orbits for G on I. For each 1 k n let i k be an element of I k. Then n I = I k = G/Stab G (i k ) i=1 n i=1 Definition Let G be a finite group and p a prime. A p-subgroup of G is a subgroup of G which is a p-group. A Sylow p-subgroup of G is a maximal p-subgroup of G, that is S is a Sylow p-subgroup of G provided that (i) S is a p-subgroup of G. (ii) If P is a p-subgroup of G with S P, then S = P. Syl p (G) denotes the set of Sylow p-subgroups of G. Fixed-Point Formula Let p be a prime and P a p-group acting on finite set I. Then In particular, if p I, then P has a fixed-point on I. I Fix I (P ) (mod p) Theorem Let G be a finite group and p a prime. 3
4 (a) (Second Sylow Theorem) G acts transitively on Syl p (G) by conjugation, that is any two Sylow p- subgroups of G are conjugate in G and so if S and T are Sylow p-subgroups of G, then S = gt g 1 for some g G. (b) (Third Sylow Theorem) p. The number of Sylow p-subgroups of G divides G and is congruent to 1 modulo (c) Let S Syl p (G). Then Syl p (G) = G/N G (S). First Sylow Theorem Let G be a finite group, p a prime and S Syl p (G). Let G = p k l with k N, l Z + and p l (p k is called the p-part of G ). Then S = p k. In particular, Syl p (G) = {P G P = p k } and G has a subgroup of order p k. Corollary and Let F K and K E be finite field extensions. Then also F E is a finite field extension dim F E = dim F K dim K E. Theorem Let F K be a field extension and a K. Suppose that a is algebraic over F. Then (a) There exists a unique monic polynomial p a F[x] with ker φ a = (p a ). (b) φ a F[x]/(p a ) F[a], f + (p a ) f(a) is a well-defined isomorphism of rings. (c) p a is irreducible. (d) F[a] is a subfield of K. (e) Let n = deg p a. Then (1, a,..., a n 1 ) is an F-basis for F[a] (f) F F[a] is finite and dim F F[a] = deg p a. (g) Let g F[x]. Then g(a) = 0 K if and only if p a g in F[x]. Proposition Let F be a field and f F[x]. Then there exists a splitting field K for f over F. Moreover, F K is finite of degree at most n!. Corollary Let F be a field, f F[x] and let K, K 1, K 2 be splitting fields of f over F. (a) There exists a field isomorphism ρ K 1 K 2 with ρ F = id F.. (b) Let p be an irreducible divisor of f in F[x] and let a 1 and a 2 be roots of p in K. Then there exists a field isomorphisms ρ K K with ρ F = id F and σ(a 1 ) = a 2. Proposition Let F K be a field extension and let G a finite subgroup of Aut F (K). Suppose that Fix K (G) = F and let a K. a 1, a 2,... a n be the distinct elements of Ga = {σ(a) σ G}. (a) a is algebraic over F. 4
5 (b) p a = (x a 1 )(x a 2 )... (x a n ). (c) p a splits over K. (d) F K is separable. Theorem Let F K be a field extension. Then the following statements are equivalent. (a) K is the splitting field of a separable polynomial over F. (b) Aut K (F) is finite and F = Fix K (Aut F (K)). (c) F = Fix K (G) for some finite subgroups G of Aut F (K). (d) F K is finite, separable and normal. Theorem Fundamental Theorem of Galois Theory intermediate field of F K and G Aut F (K). Let F K be a Galois extension. Let E be an (a) The function E Aut E (K) is a bijection between to intermediate field of K F and the subgroups of Aut F (K). The inverse of this function is given by (b) G = dim FixK (G) K and dim E K = Aut E (K). G Fix K (G). (c) F E is normal if and only if Aut E (K) is normal in Aut F (K). (d) If F E is normal, then the restriction function is a well-defined isomorphism of groups. Aut F (K)/Aut E (K) Aut F (E), σaut( F (E) σ E Part II Look at all your homeworks and compare your answers with the solutions I handed out. Pay close attention to the format of the proofs. (If you need one of the solution sets, they are available on my webpage: meier). Part III Here are some exercises similar to the ones on the upcoming exam. 1. Let G be a group and N a normal subgroup of G with G/N Sym(4). Show that there exists H G with G/H Sym(3). (Hint: Use Example and the Third Isomorphism Theorem) 2. Let P be group of order 25 acting on a set I with I = 67. Show that Fix I (P ) 2. 5
6 3. Let G be a group acting transitively on a set I. Suppose G = 60 and I = 4. Let φ be the homomorphism associated to the action of G on I. Show that Kerφ = 5 or 15. (Hints: Let i I. Show that G/Stab G (i) = 4 and Kerφ Stab G (i). Use the First Isomorphism Theorem. What can you say about Im φ?) 4. Let A and B be normal subgroups of the finite group G. Suppose that G/A = 12 and G/B = 35. Show that G = AB. (Hint: Show that G/AB divides G/A and G/B.) 5. Let G be a group of order Show that G has a unique subgroup of order 31 and a unique subgroup of order Let G be a simple group of order Show that G has fourteen Sylow 13-subgroups and that G is isomorphic to a subgroup of Sym(14). 7. Let G be a group of order Let A, B and C be subgroups of G with A = 5, B = 25 and C = 25. (a) Show that there exists g G with C = gbg 1. (b) Show that there exists h G with A hbh Let G be a finite group, p a prime, S a Sylow p-subgroup of G and P a normal p-subgroup of G. Show that P S. 9. Find a Sylow 7-subgroup of Sym(7). 10. Let F K be a Galois extension with dim F K = 20. Show that there exists an intermediate field E of F K with dim E K = Let F K be a Galois extension with Aut F (K) Sym(4). Show that there exists an intermediate field E of F K such that F E is Galois and Aut F (E) Sym(3). (Hint: Use Example and the Fundamental Theorem Of Galois Theory) 12. Let F = Z 3, let F K a field extension and let a K with K = F[a]. Suppose that a 2 = 1. Show that (a) x is the minimal polynomial of a over F. (b) Show that for each k K there exist unique c, d F with k = c + da. (c) Show that K = (a) Show that x 2 7 is an irreducible and separable polynomial over Q. (b) Show that Q[ 7] is a splitting field of x 2 7 over Q. (c) Show that there exists σ Aut Q Q[ 7] with σ( 7) = 7. (d) Show that σ(a + b 7) = a b 7 for all a, b Q. 14. Let G be a finite group and N G. Suppose that N = 12 and G/N = 75. Show that G has a subgroup of order Let F K be a Galois extension and E an intermediate field of F K. Suppose dim F K = 56 and dim E K = 7. Suppose that F E is not normal. Show that there exist exactly eight intermediate fields L of F K with dim L K = 7. Moreover, for any such L there exists σ Aut K (F) with σ(e) = L. (Hint: Use the Sylow Theorems, Lemma and the Fundamental Theorem of Galois Theory) 6
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