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
Lagrange’s Theorem Section 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).
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.
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.
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.
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 (1736-1813) was a great Italian mathematician who made fundamental contributions to all the areas of mathematics of his day.
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.
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.
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.
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.
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 = a1a2an must satisfy x2 = e. If p is a prime number of the form 4n + 3, show that we cannot solve x2 -1 mod p.
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.
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