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Groups Definition: A group is a non-empty set G equipped with a binary operation * that satisfies the following axioms: 1. Closure: If a ∈ G and b ∈ G , then a ∗ b ∈ G . 2. Associativity: a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c for all a, b, c ∈ G . 3. There is an element e ∈ G (called the identity) such that a ∗ e = e ∗ a = a for every a ∈ G. 4. For each a ∈ G , there is an element d ∈ G (called the inverse of a) such that a ∗ d = e = d ∗ a. A group is said to be abelian if it also satisfies 5. Commutativity: a ∗ b = b ∗ a for all a, b ∈ G . A group is said to be finite (or of finite order) if it has a finite number of elements. In this case the number of elements in G is called the order of G and is denoted by o(G) or |G|. A group with infinitely many elements is said to have infinite order. We have considered the following examples of finite groups: 1. Dn , (the dihedral groups of order 2n) corresponding to the groups of symmetries of the regular n-gons. 2. Sn , (the permutation groups of order n!) corresponding to the group of bijections of the set {1, 2, 3, …, n} to itself. 3. Zn , (the set of integers modulo n) composed of the equivalence classes of integers modulo n, under the operation of addition. Problem 1: Give an example of an infinite abelian group as well as an infinite nonabelian group. Theorem: Let G and H be groups. Define G × H to be the set of ordered pairs (g,h) where g ∈ G and h ∈ H . Define an operation • on G × H by ( g, h ) • ( g′, h ′) = ( g ∗ g′, h ∗ h ′) . Problem 2: (a) Show that G × H is a group. (b) Show that if G and H are both abelian, then so is G × H . Problem 3: Give examples of nonabelian groups of orders 12, 16, 30, and 48. Problem 4: If G is a group under the stated operation, prove it; if not, give a conterexample: (a) G is the set of rational numbers Q; a ∗ b = a + b + 3 ab (b) G is the set of non-zero rational numbers; a ∗ b = 3 (c) G=GL(2,R), the set of 2 × 2 invertible matrices with real entries; the operation is the usual matrix product. We will prove the following in class: Theorem: Let G be a group and let a, b, c ∈ G . Then (1) G has a unique identity element. (2) Cancellation holds in G: If a ∗ b = a ∗ c , then b = c . If b ∗ a = c ∗ a , then b = c . (3) Each element of G has a unique inverse. Since inverses are unique, we can denote the inverse of a by a−1 . Problem 5: Let G be a group and let a, b, c ∈ G . Prove (1) ( ab )−1 = b−1a−1. (2) ( a−1 )−1 = a . Definition: An element a in a group is said to have finite order if a k = e for some positive integer k. In this case, the order of the element a is the smallest positive integer n for which a n = e . The order of a is denoted by o(a) or |a|. An element a is said to have infinite order if a k ≠ e for every positive integer k. Examples: (a) In the group of nonzero rationals under multiplication, every element other than 1 and –1 has infinite order. (b) In the group of integers modulo n under addition, every element has finite order. Problem 6: (a) GL(2,R) is the set of 2 × 2 invertible matrices with real entries, as in problem 4(c). 0 1 0 −1 has order 3 and B = has order 4. Show that A = −1 −1 1 0 (b) Show that AB has infinite order. We will prove the following in class: Theorem: Let G be a group and a ∈ G . (1) If a has infinite order, then the elements a k are distinct (k an integer.) (2) If a has finite order n, then a k = e if and only if n|k , and a i = a j if and only if i ≡ j (mod n). (3) If a has order n and n=td with d>0, then a t has order d. Problem 7: Let G be a group and let a ∈ G . If a i = a j with i ≠ j , then a has finite order. Problem 8: True or false: A group of order n contains and element of order n. Justify your answer.