ProblemSet3

# ProblemSet3 - G of order 2 is odd In particular G must...

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Homework 3 Due: Wednesday 2/3/10 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. Classify each of the following as a group or not. For each which is a group give the identity and state if it is abelian or not. For each which is not a group explain which of the requirements fail. (a) ( Z , - ) (b) (GL 2 ( R ) , · ) (c) (GL 2 ( Z ) , · ) (d) ( Z 5 \ { 0 } , · mod 5) (e) ( Z 6 \ { 0 } , · mod 6) (f) The set { 5 , 15 , 25 , 35 } under multiplication modulo 40. ( Note: Z n = the set of congruence classes of the integers modulo n ) 2. Suppose the following is a Cayley table for a group. Fill in the blank entries and state whether or not the group is abelian. * e a b c d e e a b e b c d e c d a b d 3. Prove that if G is a group with the property that the square of every element is the identity then G is abelian. 4. If G is a group with an even number of elements, prove that the number of elements in
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Unformatted text preview: G of order 2 is odd. In particular, G must contain an element of order 2. 5. For x an element of G , where G is a group, show that x and x-1 have the same order. 6. Find elements A , B and C in D 4 (the group of symmetries of the square) such that AB = BC but A 6 = C . 7. Assume H is a nonempty subset of ( G,? ) which is closed under the binary operation on G and is closed under inverses, i.e., for all h and k ∈ H , hk and h-1 ∈ H . Prove that H is a group under the operation ? restricted to H . ( H is called a subgroup of G ) 8. Let G be a group, and let y ∈ G have order m . If m = dt for some d ≥ 1, prove that y t has order d ....
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