prelim2 - Spring 2010 MATH 4320: Prelim 2 Instructor: Yuri...

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Unformatted text preview: Spring 2010 MATH 4320: Prelim 2 Instructor: Yuri Berest The exam is due Wednesday, April 14 . Please write clearly and concisely. Problem 1. (15 points) If H 1 and H 2 are two groups, define their direct product H 1 H 2 to be the set of ordered pairs { ( x 1 , x 2 ) : x 1 H 1 , x 2 H 2 } equipped with the operation ( x 1 , x 2 ) ( y 1 , y 2 ) = ( x 1 y 1 , x 2 y 2 ) . ( a ) Show that the natural maps 1 : H 1 H 2 H 1 , ( x 1 , x 2 ) 7 x 1 , and 2 : H 1 H 2 H 2 , ( x 1 , x 2 ) 7 x 2 , are surjective group homomorphisms. What are the kernels of 1 and 2 ? ( b ) Prove the following property of ( H 1 H 2 , 1 , 2 ) : given any group G together with two homomorphisms f 1 : G H 1 and f 2 : G H 2 , there is a unique homomorphism : G H 1 H 2 such that f 1 = 1 and f 2 = 2 . Problem 2. (20 points) Let N and H be subgroups of a group G with N normal. Let Ad g : G G , x 7 gxg- 1 , be conjugation by an element...
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This note was uploaded on 06/07/2010 for the course MATH 201 taught by Professor Crissinger during the Spring '08 term at University of Delaware.

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prelim2 - Spring 2010 MATH 4320: Prelim 2 Instructor: Yuri...

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