practice2c

# practice2c - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS February 24, 2007 Here are a few practice problems on groups. Try to work these WITHOUT looking at the solutions! After you write your own solution, you can compare to my solution. Your solution does not need to be identical—there are often many ways to solve a problem—but it does need to be CORRECT. 1. Suppose that N is a normal subgroup of a group G . Show that if a G has ±nite order o ( a ), then Na in G/N has ±nite order m with m | o ( a ). Solution Let n = o ( a ). Then a n = e , so ( Na ) n = Na n = Ne = N , which is the identity element of G/N . Hence we must have o ( Na ) | n . ± Question: Will it be true that o ( Na ) = o ( a )? (No.) Try an example. 2. a. Suppose that H and K are groups. I won’t prove it here, but you should be able to prove that the Cartesian product H × K = { ( h, k ) : h H, k K } is a group with group operation ( h 1 , k 1 )( h 2 , k 2 ) = ( h 1 h 2 , k 1 k 2 ). Prove the following facts about this group. a. Show that

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practice2c - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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