Conjugate Elements of a Symmetric Group

  • #1
Is the the following definition correct?

Two elements a and b of a group G are said to be CONJUGATE if there exists g in G such that [tex]a=gbg^{-1}[/tex].

For instance, show that all elements in the symmetric group S5 of order 6 conjugate.
 

Answers and Replies

  • #2
matt grime
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EDIT: Yes, that's the correct definition, as well.


Again, do it: take an element of order 6, compute its conjugates with a couple of elements and see who to generate all elements of order 6.

Eg, in S_n n>2, consider (12)(23)(12)=(13), thus it's clear that all elements of order 2 are conjugate
 
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  • #3
matt grime
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Incidentally, when i say "just do it" i'm not being impatient, it's just that sometimes in maths you sadly just have to get your hands dirty with some calculations.
 
  • #4
I know, I just think I remember a theorem that says two elements in a symmetric group are conjugate if and only if they have the same cycle shape. There are 7 different cycle shapes in S5, I think.
 
  • #5
matt grime
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Not sure what you're getting at. Perhaps if you realized there was only one cycle shape which corresponded to elements of order 6 that would help.
 

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