m321s1

m321s1 - B U Department of Mathematics Fall 2008 Math 321 -...

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Unformatted text preview: B U Department of Mathematics Fall 2008 Math 321 - Algebra, First Midterm Exam, 14/11/2008, 17:00-19:00 I hope you enjoy the exam! Full Name Student ID : : over 100 In what follows, Zn is the additive group with the modular addition; G, H and K are groups. 1. Solve the following questions within the space allocated. Do not write in the margins or wheresoever. Expain your answer. These are not just yes/no questions. Figures are from Wikipedia. (a) [10pts] Let f : G → H and g : H → K be isomorphisms. Is it true that g ◦ f is an isomorphism? (b) [10pts] Give an example of two non-isomorphic groups of the same order. Explain. (c) [10pts] Find all subgroups of Z15 . Explain why. (d) [5pts] What is the order of the group of symmetries of a cube? (e) [5pts] What is the order of the group of symmetries of a dodecahedron? 2. We define an operation · on G × H = {(g, h)|g ∈ G, h ∈ H } as follows: . (g1 , h1 ) · (g2 , h2 ) = (g1 g2 , h1 h2 ). (a) [10pts] Show that (G × H, ·) is a group. (b) [5pts] Let g ∈ G and h ∈ H have orders m and n respectively. What is the order of (g, h) in G × H ? (c) [5pts] Is Z2 × Z3 isomorphic to Z6 ? (d) [10pts] Let pG be the projection map from G × H to G, i.e. pG (g, h) = g . For arbitrary subgroup L of G × H , is it true that pG (L) is a subgroup of G? If yes prove, if no give a counter example. 3. The subgroup N is normal in G if and only if for every g ∈ G and h ∈ N , the conjugate ghg−1 stays in N . In S5 , by a 5-cycle we mean permutations of the form (a1 a2 a3 a4 a5 ); 3-cycles are of the form (a1 a2 a3 ); (2-2)-cycles are of the form (a1 a2 )(a3 a4 ). Here ai ’s are distinct integers between 1 and 5. (a) [5pts] In S5 calculate: (12345) · (13245) = (123) · (145) = (12)(34) · (12)(45) = (b) [15pts] Let N be a normal subgroup of A5 . If N contains a 5-cycle, prove that N contains every other 5-cycle. Same question for 3-cycles and for (2-2)-cycles. (c) [10pts] Using parts (a) and (b), prove that the only normal subgroups of A5 are A5 and {1}. ...
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m321s1 - B U Department of Mathematics Fall 2008 Math 321 -...

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