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Notes on Math 571 (Abstract Algebra), Chapter Iii: Quotient Groups and Homomorphisms

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Notes on Math 571 (Abstract Algebra), Chapter Iii: Quotient Groups and Homomorphisms NOTES ON MATH 571 (ABSTRACT ALGEBRA), CHAPTER III: QUOTIENT GROUPS AND HOMOMORPHISMS LI LI 3.1 Definitions and examples Start from the example Z ! Z=nZ, point out the difference to the notion of a subgroup: there is no nontrivial homomorphism Z=nZ ! Z. Explain the notion “fiber". Proposition 1. Let ' : G ! H be a homomorphism of groups. (1) '(1G) = 1H . (2) '(g−1) = '(g)−1 for all g 2 G. n n (3) '(g ) = '(g) for all g 2 G; n 2 Z. (4) ker' is a subgroup of G. (5) im' is a subgroup of H. Definition 2. Let ' : G ! H be a homomorphism with kernel K. The quotient group or factor group, G=K, is the group whose elements are the fibers of ', and the group operation is: if X = '−1(a) and Y = '−1(b), then XY is defined to be '−1(ab). Remark. G=K is \the same" as im'. Definition 3. For N ≤ G and g 2 G, define gN = fgnjn 2 Ng; Ng = fngjn 2 Ng called a left coset and a right coset of N in G. Any element in a coset is called a representative. Remark. For additive group, cosets are written as g + N or N + g. Proposition 4. Let ' : G ! H be a group homomorphism and K be the kernel. Let a be an element in im' and X = '−1(a). Then (1) For any u 2 X, X = uK. (2) For any u 2 X, X = Ku. Remark. uK = Ku, i.e. uKu−1 = K. Definition 5. The element gng−1 is called the conjugate of of n by g. The set gNg−1 is called the conjugate of N by g. The element g is said to normalize N if gNg−1 = N. −1 A subgroup N of a group G is normal, denoted by N E G, if gNg = N for all a 2 G. 1 2 LI LI Kernel of a group homomorphism is a normal subgroup. On the other hand, we can always find a group homomorphism whose kernel is a pre- scribed normal subgroup. To see that, let us first be familiar with several equivalent descriptions of a normal subgroup. Theorem 6. Let N be a subgroup of G. TFAE (1) N E G (2) NG(N) = G (3) gN = Ng for all g 2 N (4) the operation on left cosets defined by uN · vN = uvN makes the set of left cosets into a group (5) gNg−1 ⊆ N for all g 2 N Proposition 7. A subgroup N of G is normal iff it is the kernel of some homomorphism. Exercise on class: × × 2 2 Example. Define C ! R by '(a+bi) = a +b . Show that ' is a morphism and find the image of '. Describe the kernel and fibers of ' geometrically. Example. Any subgroup of an abelian group is normal. Example. The center of a group is normal. Example. Show that any quotient group of a cyclic group are cyclic. Example. h(123)i E S3. Example. Let D2n be the dihedral group. (1) Prove that hri is a normal subgroup of D2n and describe D2n=hri. ∼ (= Z=2Z.) k ∼ k (2) Let kjn, describe D2n=hr i.(= D2k. Construct the map D2n=hr i ! D2k, show it is a group isomorphism.) 3.2 More on cosets and Lagrange's Theorem Recall Lagrange's Theorem. If H ≤ G, the number of left coset of H in G is jGj=jHj. (also recall the proof: jgHj = jHj, and G is partitioned into disjoint subsets, each of which has n elements.) Definition 8. Given H ≤ G, the number of left cosets of H in G is called the index of H in G, denoted by jG : Hj. Example. jZ : 2Zj = 2. Two consequence of Lagrange's Theorem: Corollary 9. If G is a finite group and x 2 G, the order of x divides the order of G. (Consider the subgroup hxi.) ∼ Corollary 10. If jGj = p is prime, then G = Z=pZ. NOTES ON MATH 571 (ABSTRACT ALGEBRA), CHAPTER III: QUOTIENT GROUPS AND HOMOMORPHISMS3 The reverse of Lagrange's theorem is not true in general, but we have the following two theorem, whose proof will be postponed. Theorem 11. (Cauchy's theorem) If G is a finite group and p is a prime number dividing jGj. Then G has an element of order p. Theorem 12. (Sylow) If G is a finite group of order pαm, where p is a prime and pP does not divide m, then G has a subgroup of order pα. Now we mention some important facts involving cosets. Definition 13. Given H; K subgroups of a given group, define HK = fhkjh 2 H; k 2 Kg: Remark. HK may not be a subgroup. It is a subgroup iff HK = KH. Proposition 14. If H an K are subgroups of a group, HK is a subgroup iff HK = KH. (Two directions. \(" can be verified by subgroup criterion. \)": to show KH ⊆ HK is straightforward; to show HK ⊂ KH we use a trick. −1 Take hk 2 HK, write hk = (h1k1) .) Corollary 15. If H and K are subgroups of G and H ≤ NG(K), then HK is a subgroup of G. ( Since any h 2 H commutes with K, the subgroup H commutes with K. THen by the above proposition HK is a subgroup.) Example. Fermat's Little Theorem: if p is a prime then ap = a(mod p) for all a 2 Z. Proposition 16. If H and K are finite subgroups of a group, then jHjjKj jHKj = : jH \ Kj (Decompose HK as union of hK. Show that h1K = h2K iff h1(H \K) = h2(H \ K). Therefore the number of distinct cosets of the form hK is equal to the number of cosets of H \ K in H.) 3.3 The isomorphism Theorems Theorem 17. (The first isomorphism theorem) If ' : G ! H is a homo- ∼ morphism of groups. Then ker' E G and G=ker' = '(G). (Recall the definition of quotient group. Each coset gK sends to '(g). Then ' : G ! '(G) is a group homomorphism and is surjective. Then show it is injective.) Theorem 18. (The second isomorphism theorem) Let G be a group. Let A, B be subgroups of G and assume A ≤ NG(B). Then AB is a subgroup ∼ of G, B E AB, A \ B E A and AB=B = A=A \ B. 4 LI LI (First point out A ≤ NG(B) implies AB = BA implies AB is a subgroup. Then draw the picture G AB normal A B normal A \ B 1 −1 −1 Use abBb a = B to prove that B E AB. Construct the group homomorphism ' : A ! AB=B by '(a) = aB. Point out the group operation on AB=B is well defined since B is a normal subgroup of AB. It follows that ' is a group homomorphism. Surjectivity follows from definition of AB. Compute the kernel of ', which is A \ B. Use First Isomorphism Theorem to complete the proof. ) Theorem 19. (The third isomorphism theorem) Let G be a group and let H; K be normal subgroups of G with H ≤ G. Then K=H E G=H and (G=H)=(K=H) = G=K: (It is an easy exercise that K=H is a normal subgroup of G=H. Then consider the map ' : G=H ! G=K sending gH to gK. Show that this is well-defined, is a group homomorphism, and is surjective. Then find its kernel is K=H.) 3.4 Composition Series and the H¨olderProgram Definition 20. A group is called simple if jGj > 1 and the only normal subgroup of G are 1 and G. For example, if jGj is prime, then G is simple. The smallest non-abelian simple group is A5, which has order 60. Definition 21. In a group G, a sequence of subgroups 1 = N0 ≤ N1 ≤ N2 ≤ · · · ≤ Nk−1 ≤ Nk = G is called a composition series if Ni E Ni+1 and Ni+1=Ni is a simple group for 0 ≤ i ≤ k − 1. If the above sequence is a composition series, the quotient groups Ni+1=Ni are called composition factors of G. NOTES ON MATH 571 (ABSTRACT ALGEBRA), CHAPTER III: QUOTIENT GROUPS AND HOMOMORPHISMS5 2 2 Example. 1 ≤ hsi ≤ hs; r i ≤ D8 and 1 ≤ hr i ≤ hri ≤ D8. Theorem 22. (Jordan-H¨older)Let G be a finite group with G 6= 1. Then (1) G has a composition series, and (2) The composition factors in a composition series are unique, namely, if 1 = N0 ≤ N1 ≤ · · · ≤ Nr = G 1 = M0 ≤ M1 ≤ · · · ≤ Ms = G are two composition series of G, then r = s and there is some permutation π of 1:::r such that Mπ(i)=Mπ(i)−1 = Ni=Ni−1..
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