2
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Does this look right.....?

Find all numbers $n$ such that $S_5$ contains an element of order $n$.

Because any element in $S_5$ has order $n$ which is the lcm of the order all of its cycle decomposition. Clearly, we can make cycle decompositions with

  • Order 1: The identity.

  • Order 2: (12)

  • Order 3: (124)

  • Order 4: (1234)

  • Order 5: (12345)

  • Order 6: (12)(345)

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1
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You're missing at least one. Have you checked an element of every conjugacy class? Recall that conjugacy classes of $S_n$ are parameterized by partitions of $n$.

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  • $\begingroup$ Good insight. One should be $(123)(45)$, whose order is $6$. $\endgroup$ – Yai0Phah Sep 8 '13 at 4:51
  • $\begingroup$ Oh Yeah, I see the right way to see it. Thank you. $\endgroup$ – Tumbleweed Sep 8 '13 at 4:56
3
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I did the below codes using GAP to find out that $n$. We know that if $G$ wants to have an element of order $n$ so $n\mid |G|$. Here, $|G|=5!=120$, so the following numbers are its dividing factors:

$$1,~ 2,~ 3,~ 4,~ 5,~ 6,~ 8,~ 10,~ 12,~ 15,~ 20,~ 24,~ 30,~ 40,~ 60,~ 120$$

gap> e:=Elements(SymmetricGroup(IsPermGroup,5));;
     k:=DivisorsInt(120);;
     for i in k do Print("The following elements are of order","  ", "i=", 
         i,":",Filtered(e,t->Order(t)=i),"\n"); 
     od;

Output:

The following elements are of order  i=2:
[ (4,5), (3,4), (3,5), (2,3), (2,3)(4,5), (2,4), (2,4)(3,5), (2,5), 
(2,5)(3,4), (1,2), (1,2)(4,5), (1,2)(3,4), (1,2)(3,5), (1,3), (1,3)(4,5), 
(1,3)(2,4), (1,3)(2,5), (1,4), (1,4)(3,5), (1,4)(2,3), (1,4)(2,5), (1,5), 
(1,5)(3,4), (1,5)(2,3), (1,5)(2,4) ]
The following elements are of order  i=4:
[ (2,3,4,5), (2,3,5,4), (2,4,5,3), 
(2,4,3,5), (2,5,4,3), (2,5,3,4), (1,2,3,4), (1,2,3,5), (1,2,4,3), 
(1,2,4,5), (1,2,5,3), (1,2,5,4), (1,3,4,2), (1,3,5,2), (1,3,4,5), 
(1,3,5,4), (1,3,2,4), (1,3,2,5), (1,4,3,2), (1,4,5,2), (1,4,5,3), 
(1,4,3,5), (1,4,2,3), (1,4,2,5), (1,5,3,2), (1,5,4,2), (1,5,4,3), 
(1,5,3,4), (1,5,2,3), (1,5,2,4) ]
The following elements are of order  i=6:
[ (1,2)(3,4,5), (1,2)(3,5,4), (1,2,3)(4,5), (1,2,4)(3,5), (1,2,5)(3,4), 
(1,3,2)(4,5), (1,3)(2,4,5), (1,3,5)(2,4), (1,3)(2,5,4), (1,3,4)(2,5), 
(1,4,2)(3,5), (1,4,5)(2,3), (1,4)(2,3,5), (1,4,3)(2,5), (1,4)(2,5,3), 
(1,5,2)(3,4), (1,5,4)(2,3), (1,5)(2,3,4), (1,5,3)(2,4), (1,5)(2,4,3) ]

The following elements are of order  i=8:[  ]
The following elements are of order  i=10:[  ]
The following elements are of order  i=12:[  ]
The following elements are of order  i=15:[  ]
The following elements are of order  i=20:[  ]
The following elements are of order  i=24:[  ]
The following elements are of order  i=30:[  ]
The following elements are of order  i=40:[  ]
The following elements are of order  i=60:[  ]
The following elements are of order  i=120:[  ]
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  • $\begingroup$ wow, magic Babak! $\endgroup$ – Tumbleweed Sep 10 '13 at 7:33
  • $\begingroup$ @Tumbleweed: (-: $\endgroup$ – mrs Sep 10 '13 at 7:35
  • $\begingroup$ Magic indeed! +1 $\endgroup$ – Namaste Sep 10 '13 at 14:03

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