1. [9 points] The set  G= x x x x   : x ∈ R − {0} is a group under multiplication.   1/2 1/2 (a) Show that is an identity element for G. 1/2 1/2  (b) Find the inverse of x x x x  in G. Show your work. (c) Is the following claim true or false? Explain your answer.   x x 7→ x is a homomorphism. The map φ : G → (R − {0}, ·), x x 2 2. [8 points] (a) Find a subgroup of Z30 of order 6. (No need to show work.) (b) Find a subgroup of Z8 × Z2 of index 2. (No need to show work.) (c) Give an example of a group G and a subset H ⊆ H which is closed under the operation, but is not a subgroup. Explain your answer. (d) Give an example of a group G and a subset H ⊆ H which contains the inverse of all its elements, but is not a group. (No need to show work. ) 3 3. [12 points] Consider the group U (24) = {[a]24 : GCD(a, 24) = 1} under multiplication modulo 24. (a) List the elements of U (24) and find their orders. (b) Prove or disprove: U (24) is isomorphic to D4 . 4 (c) Prove or disprove: U (24) is isomorphic to Z4 × Z2 . (d) Prove or disprove: U (24) is isomorphic to a subgroup of a cyclic group. (e) Find a familiar group isomorphic to U (24). Explain your answer. 5 4. [10 points] (a) List - up to isomorphism - all abelian groups of order 100. (Avoid repetitions.) (b) Which of these groups contains exactly 3 elements of order 2, and 4 elements of order 5? Explain your answer, and explicitly list the elements of order 2 and 5 in the chosen group. 6 5. [8 points] Give an example or prove why none exists: (a) An abelian group G with a non-abelian subgroup (b) A group G whose proper subgroups are all abelian, but G is not abelian (c) An abelian group G with a non-abelian factor group G/H (d) A non-abelian group G with an abelian factor group G/H 7 6. [6 points] Let F be the set of all (continuous and) differentiable functions. Which of the following is a homomorphism of binary structures? Briefly explain your answer. (a) ψ : (F, +) → (R, +), f 7→ f 0 (0) (the operation on F is point-wise addition of functions) (b) ψ : (F, ·) → (R, ·), f 7→ f 0 (0) (the operation on F is point-wise multiplication of functions) 8 7. [10 points] Let G = Z4 × Z4 , H = h([2]4 , [3]4 )i. (a) Find a, b, c, d ∈ G so that G is the disjoint union of the 4 cosets a + H, b + H, c + H, d + H. List the elements of each coset. (b) Is G/H cyclic? Explain your answer. 9 8. [6 points] Let G = S9 . (a) Decompose (643)(17)(4157) as a product of disjoint cycles. No need to show work. (b) Decompose (643)(17)(4157) as a product of transpositions. No need to show work. (c) Find the inverse of (643)(17)(4157).No need to show work. (d) Find the order of (643)(17)(4157). No need to show work. 9. [10 points] Every element of S6 can be written as a product of disjoint cycles. The possible cyclic structures are listed below: • ( ) • ( ) • ( )( ) • ( ) • ( )( • ( )( ) • ( ) ) • ( )( )( ) • ( )( ) • ( ) • ( )( )( )( )( )( ). 10 (a) List the possible cyclic structures in A6 . Explain your reasoning. (b) Now use this information to determine all possible orders in A6 . Show our work. 11 10. [6 points] Let G be a group of order 91 (= 13 × 7), not necessarily cyclic. Use Lagrange’s theorem to prove that every proper subgroup of G is cyclic. 12 11. [15 points] You only have to complete 3 out of the following 4 parts. Clearly indicate which three parts you want to be graded or we will pick the first three by default. For each part you select, give a counterexample for the false claim that is provided, and explain your answer. Then prove that the corresponding “fixed statement” is true. (a) False Claim: Let H be a subgroup of a group G and let a belong to G. Then aH = Ha. B Counterexample: Fixed statement: Let H be a subgroup of a group G and let a belong to G. Assume that aha−1 ∈ H, for all a ∈ G and for all h ∈ H. Then aH = Ha. B Proof: 13 (b) False Claim: Let G and H be cyclic group. Then the group G × H is cyclic. B Counterexample: Fixed statement: Let G and H be cyclic group. Assume that G and H are finite, and that GCD(|G|, |H|) = 1. Then the group G × H is cyclic. B Proof: 14 (c) False Claim: Let G be a finite group and let d be a positive integer that divides |G|. Then G contains an element of order d. B Counterexample: Fixed statement: Let G be a finite group and let d be a positive integer that divides |G|. Assume that G is cyclic. Then G contains an element of order d. B Proof: 15 (d) False Claim: Let G be a group and let H = {a ∈ G : a2 = e} be the subset of G containing all the elements of order at most 2. Then H is a subgroup of G. B Counterexample: Fixed statement:Let G be a group and let H = {a ∈ G : a2 = e} be the subset of G containing all the elements of order at most 2. Assume G is abelian. Then H is a subgroup of G. B Proof: 16
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