Garis-garis Besar Perkuliahan

1. Garis-garis Besar Perkuliahan 15/2/10 Sets and Relations 22/2/10 Definitions and Examples of Groups 01/2/10 Subgroups 08/3/10 Lagrange’s Theorem 15/3/10 Mid-test 1 22/3/10 Homomorphisms and Normal Subgroups 1 29/3/10 Homomorphisms and Normal Subgroups 2 05/4/10 Factor Groups 1 12/4/10 Factor Groups 2 19/4/10 Mid-test 2 26/4/10 Cauchy’s Theorem 1 03/5/10 Cauchy’s Theorem 2 10/5/10 The Symmetric Group 1 17/5/10 The Symmetric Group 2 22/5/10 Final-exam

2. Factor Groupsand Cauchy’s Theorem

3. Theorem 1 If NG and G/N = {Na | a  G}, then G/N is a group under the operation (Na)(Nb) = Nab. If G is a finite group and NG, then |G/N| = |G|/|N|.

4. Theorem 2 If G is a finite abelian group of order |G| and p is a prime that divides |G|, then G has an element of order p.

5. Problems • If G is a cyclic group and N is a subgroup of G, show that G/N is a cyclic group. • If G is an abellian group and N is a subgroup of G, show that G/N is an abelian group. • Let G be an abelian group of order mn, where m and n are relatively prime. Let M = {a  G| am = e}. Prove that: • M is a subgroup of G. • G/M has no element, x, other than the identity element, such that xm = unit element of G/M.

6. Theorem 3 First Homomorphism Theorem Let  be a homomorphism of G onto G’ with kernel K. Then G’  G/K, the isomorphism between these being effected by the map  : G/K  G’ defined by (Ka) = (a).

7. Theorem 4 Correspondence Theorem Let  be a homomorphism of G onto G’ with kernel K. If H’ is a subgroup of G’ and if H = {a  G | (a)  H’}, then H is a subgroup of G, K  H, and H/K  H’. Finally, if H’  G’, then H  G.

8. Theorem 5 Second Homomorphism Theorem Let H be a subgroup of a group G and N a normal subgroup of G. Then HN = {hn| h  H, n  N} is a subgroup of G, HN is a normal subgroup of H, and H/(HN)  (HN)/N.

9. Theorem 6 Third Homomorphism Theorem If  is a homomorphism of G onto G’ with kernel K, then, if N’  G’ and N = {a  G | (a)  N’}, we conclude that G/N  G’/N’. Equivalently, G/N  (G/K)/(N/K).

10. Problems • Let G be the group of all real-valued functions on the unit interval [0,1], where we define, for f, g  G, addition by (f+g)(x) = f(x)+g(x) for every x [0,1]. If N = {f  G|f()=0}, prove that G/N  real numbers under +. • If G1, G2 are two groups and G = G1  G2 = {(a,b)|a  G1, b  G2}, where we define (a,b)(c,d) = (ac,bd), show that: • N = {(a,e2)|a  G1}, where e2 is the unit element of G2, is a normal subgroup of G. • N  G1. • G/N  G2.

11. Cauchy’s Theorem Orbit Let S be a set, f A(S), and define a relation on S as follows: s  t if t = f i (s) for some integer i. Verify that this defines an equivalence relation on S. The equivalence class of s, [s], is called the orbit of s under f.

12. Cauchy’s Theorem Lemma 7 If f A(S) is of order p, p a prime, then the orbit of any element of S under f has 1 or p elements.

13. Cauchy’s Theorem Theorem 8 Ifp is a prime and p divides the order of G, then G contains an element of order p.

14. Cauchy’s Theorem Lemma 9 Let G be a group of order pq, wherep,q are primes and p > q. If a  G is of order p and A is the subgroup of G generated by a, then AG.

15. Cauchy’s Theorem Corollary 10 If G,a are as in Lemma 9 and x  G, then x-1ax = ai, for some i where 0 < i < p (depending on x)

16. Cauchy’s Theorem Lemma 11 If a  G is of order m and b  G is of order n, where m and n are relatively prime and ab = ba, c = ab is of order mn.

17. Cauchy’s Theorem Theorem 12 LetG be a group of order pq, where p,q are primes and p > q. If q p - 1, then G must be cyclic.

18. Problems • Prove that a group of order 35 is cyclic. • Construct a nonabelian group of order 21. (Hint: Assume that a3 = e, b7 = e and find some i such that a-1ba = ai ≠ a, which is consistent with the relations a3 = b7 = e.) • Let G be a group of order pnm, where p is prime and p m. Supposse that G has a normal subgroup of order pn. Prove that (P) = P for every automorphism  of G.

19. Question? If you are confused like this kitty is, please ask questions =(^ y ^)=