Conjugate Elements of a Symmetric Group

[Treadstone 71]

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 $$a=gbg^{-1}$$.

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

 

[matt grime]

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

 

[matt grime]

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.

 

[Treadstone 71]

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.

 

[matt grime]

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.