practice3a - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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Unformatted text preview: MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS March 2, 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 identicalthere are often many ways to solve a problembut it does need to be CORRECT. 1. Suppose that H is a subgroup of S n and H contains every 2-cycle ( i 1 i 2 ). Show that H = S n . In other words, we say that the 2-cycles generate S n . Hint: Show first that H contains every 3-cycle, then every 4-cycle, etc., up to the n-cycles. Then show that H contains every permutation, whether it is a cycle or not. Solution Every 3-cycle is a product of 2-cycles, because ( i 1 i 2 i 3 ) = ( i 1 i 3 )( i 1 i 2 ). Since H contains every 2-cycle and is closed under composition, it therefore contains every 3-cycle. Every 4-cycle is a product of a 3-cycle and a 2-cycle, because ( i 1 i 2 i 3 i 4 ) = ( i 1 i 4 )( i 1 i 2 i 3 )....
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practice3a - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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