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. Lagrange’s Theorem Section 3

3. Equivalence Relation
Definition. A relation ~ on a set S is called an equivalence relation if, for all a, b, c  S, it satisfies:
a ~ a (reflexivity).
a ~ b implies that b ~ a (symmetry).
a ~ b, b ~ c implies that a ~ c (transitivity).

4. Examples
Let n > 1 be a fixed integer. Define a ~ b for a, b  Z if n | (b – a). When a ~ b, we write this as a  b mod n, which is read “a congruent to b mod n.”
Let G be a group and H a subgroup of G. Define a ~ b for a, b  G if ab-1  H.
Let G be any group. Declare that a ~ b if there exists an x  G such that b = x-1ax.

5. Equivalence Class
Definition. If ~ is an equivalence relation on S, then the class of a, is defined by [a] = {b  S | b ~ a}. In Example 2, b ~ a  ba -1 H  ba -1 = h for some h  H. That is, b ~ a  b = ha  Ha = {ha | h  H}. Thus, [a] = Ha. The set Ha is called a right coset of H in G.

6. Equivalence Class
Theorem 1. If ~ is an equivalence relation on S, then S =  [a], where this union runs over one element from each class, and where [a]  [b] implies that [a]  [b] = . That is, ~ partitions S into equivalence classes.

7. Lagrange’s Theorem
Theorem 2. If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. J. L. Lagrange ( ) was a great Italian mathematician who made fundamental contributions to all the areas of mathematics of his day.

8. Order of an element
Definition. If G is finite, then the order of a, written o(a), is the least positive integer m such that am = e. Theorem 4. If G is finite and a  G, then o(a) | |G|. Corollary. If G is a finite group of order n, then an = e for all a  G.

9. Cyclic Group
A group G is said to be cyclic if there is an element a  G such that every element of G is a power of a. Theorem 3. A group G of prime order is cyclic.

10. Congruence Class mod n
Theorem 5. Zn forms a cyclic group under addition modulo n. Theorem 6. Zn* forms an abelian group under the product modulo n, of order (n). Theorem 7. If a is an integer relatively prime to n, then a(n)  1 mod n. Corollary (Fermat). If p is a prime and p  a, then ap-1  1 mod p.

11. Problems
Let G be a group and H a subgroup of G. Define a ~ b for a, b  G if a-1b  H. Prove that this defines an equivalence relation on G, and show that [a] = aH = {ah | h  H}. The sets aH are called left cosets of H in G.
If G is S3 and H = {i, f}, where f : S  S is defined by f(x1) = x2, f(x2) = x1, f(x3) = x3, list all the right cosets of H in G and list all the left cosets of H in G.

12. Problems
If p is a prime number, show that the only solutions of x2  1 mod p are x  1 mod p and x  -1 mod p.
If G is a finite abelian group and a1, a2, an are all its elements, show that x = a1a2an must satisfy x2 = e.
If p is a prime number of the form 4n + 3, show that we cannot solve x2  -1 mod p.

13. Problems If o(a) = m and as = e, prove that m | s.
If in a group G, a5 = e and aba-1 = b2, find o(b) if b  e.
In a cyclic group of order n, show that for each positive integer m that divides n (including m = 1 and m = n) there are (m) elements of order m.

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