aug26 - Math 5125 Friday August 26 August 26 Ungraded...

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Math 5125 Friday, August 26 August 26, Ungraded Homework Exercise 4.1.4 on page 116 Let S 3 act on the set Ω of ordered pairs: { ( i , j ) | 1 i , j 3 } by σ (( i , j )) = ( σ ( i ) , σ ( j )) . Find the orbits of S 3 on Ω . For each σ S 3 ﬁnd the cycle decomposition of σ under this action (i.e., ﬁnd its cycle decomposition when σ is considered as an element of S 9 – ﬁrst ﬁx a labelling of these nine ordered pairs). For each orbit O of S 3 acting on these nine points pick some a O and ﬁnd the stabilizer of a in S 3 . There are two orbits under S 3 , namely { (1,1), (2,2), (3,3) } and { (1,2), (2,3), (3,1), (1,3), (2,1), (3,2) } . The stabilizer of (1,1) in S 3 is (2 3), and the stabilizer of (1,2) in S 3 is 1. The cycle decompositions are given as follows. (1) ( ( 1 , 1 ) ) (12) ( ( 1 , 1 )( 2 , 2 ) )( ( 1 , 2 )( 2 , 1 ) )( ( 1 , 3 )( 2 , 3 ) )( ( 2 , 3 )( 1 , 3 ) ) (23) ( ( 2 , 2 )( 3 , 3 ) )( ( 2 , 3 )( 3 , 2 ) )( ( 2 , 1 )( 3 , 1 ) )( ( 3 , 1 )( 2 , 1 ) ) (13)
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