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Abstract Algebra

Exercise 4.3.7 (Representatives of the conjugacy classes in $S_n$ for $n = 3,4,\ldots,7$)

For n = 3 , 4 , 6 and 7 make lists of the partition of n and give representatives for the corresponding classes of S n .

Answers

Representatives for the conjugacy classes of S n .

  • n = 3 :

    3 σ 1 , 1 , 1 ( ) 1 , 2 ( 1 2 ) 3 ( 1 2 3 )

  • n = 4 :

    4 σ 1 , 1 , 1 , 1 ( ) 1 , 1 , 2 ( 1 2 ) 2 , 2 ( 1 2 ) ( 3 4 ) 1 , 3 ( 1 2 3 ) 4 ( 1 2 3 4 )

  • n = 5 is treated in the text p. 127

  • n = 6 :

    6 σ 1 , 1 , 1 , 1 , 1 , 1 ( ) 1 , 1 , 1 , 1 , 2 ( 1 2 ) 1 , 1 , 2 , 2 ( 1 2 ) ( 3 4 ) 2 , 2 , 2 ( 1 2 ) ( 3 4 ) ( 5 6 ) 1 , 1 , 1 , 3 ( 1 2 3 ) 1 , 2 , 3 ( 1 2 ) ( 3 4 5 ) 3 , 3 ( 1 2 3 ) ( 4 5 6 ) 1 , 1 , 4 ( 1 2 3 4 ) 2 , 4 ( 1 2 ) ( 3 4 5 6 ) 1 , 5 ( 1 2 3 4 5 ) 6 ( 1 2 3 4 5 6 )

  • n = 7 :

    3 σ 1 , 1 , 1 , 1 , 1 , 1 , 1 ( ) 1 , 1 , 1 , 1 , 1 , 2 ( 1 2 ) 1 , 1 , 1 , 2 , 2 ( 1 2 ) ( 3 4 ) 1 , 2 , 2 , 2 ( 1 2 ) ( 3 4 ) ( 5 6 ) 1 , 1 , 1 , 1 , 3 ( 1 2 3 ) 1 , 1 , 2 , 3 ( 1 2 ) ( 3 4 5 ) 2 , 2 , 3 ( 1 2 ) ( 3 4 ) ( 5 6 7 ) 1 , 3 , 3 ( 1 2 3 ) ( 4 5 6 ) 1 , 1 , 1 , 4 ( 1 2 3 4 ) 1 , 2 , 4 ( 1 2 ) ( 3 4 5 6 ) 3 , 4 ( 1 2 3 ) ( 4 5 6 7 ) 1 , 1 , 5 ( 1 2 3 4 5 ) 2 , 5 ( 1 2 ) ( 3 4 5 6 7 ) 1 , 6 ( 1 2 3 4 5 6 ) 7 ( 1 2 3 4 5 6 7 )

With Sagemath:

sage: Partitions(7).list()

[[7],
 [6, 1],
 [5, 2],
 [5, 1, 1],
 [4, 3],
 [4, 2, 1],
 [4, 1, 1, 1],
 [3, 3, 1],
 [3, 2, 2],
 [3, 2, 1, 1],
 [3, 1, 1, 1, 1],
 [2, 2, 2, 1],
 [2, 2, 1, 1, 1],
 [2, 1, 1, 1, 1, 1],
 [1, 1, 1, 1, 1, 1, 1]]

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2026-02-03 11:11
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