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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 2cycle ( i 1 i 2 ). Show that H = S n . In other words, we say that the 2cycles generate S n . Hint: Show first that H contains every 3cycle, then every 4cycle, etc., up to the ncycles. Then show that H contains every permutation, whether it is a cycle or not. Solution Every 3cycle is a product of 2cycles, because ( i 1 i 2 i 3 ) = ( i 1 i 3 )( i 1 i 2 ). Since H contains every 2cycle and is closed under composition, it therefore contains every 3cycle. Every 4cycle is a product of a 3cycle and a 2cycle, because ( i 1 i 2 i 3 i 4 ) = ( i 1 i 4 )( i 1 i 2 i 3 )....
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 Spring '10
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 Math, Algebra

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