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m321s1

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

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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, Z n 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 Z 15 . 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?

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2. We define an operation · on G × H = { ( g, h ) | g G, h H } as follows: ( g 1 , h 1 ) · ( g 2 , h 2 ) . = ( g 1 g 2 , h 1 h 2 ) . (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 Z 2 × Z 3 isomorphic to
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