# conjugacy classes

• Mar 21st 2011, 10:29 AM
poirot
conjugacy classes
question: Find the order of the folowing elements of s15 and determine which pairs are conjugate.

i) (1 2 4 8)(3 6 12)(5 10 15)

ii) (1 2)(4 8)(3 12 11)(5 13 7)

iii) (1 7 9)(11 12 13)(5 10 8 6)

I have found the orders using lcm i) 12, ii) 6, iii) 12.

To be conjugate, elements need to have same order so only i) and iii) are possibilities.
I tried using ii) as 'g' in a=gbg^-1 but it didn't work. I don't think it expects to trawl through s15 looking for a g so is there an easy way to determine conjugacy?
• Mar 21st 2011, 12:05 PM
Drexel28
Quote:

Originally Posted by poirot
question: Find the order of the folowing elements of s15 and determine which pairs are conjugate.

i) (1 2 4 8)(3 6 12)(5 10 15)

ii) (1 2)(4 8)(3 12 11)(5 13 7)

iii) (1 7 9)(11 12 13)(5 10 8 6)

I have found the orders using lcm i) 12, ii) 6, iii) 12.

To be conjugate, elements need to have same order so only i) and iii) are possibilities.
I tried using ii) as 'g' in a=gbg^-1 but it didn't work. I don't think it expects to trawl through s15 looking for a g so is there an easy way to determine conjugacy?

A possibility is to decompose them into disjoint cycles then use the fact that two elements of $S_n$ are conjugate if and only if they have the same cycle structure (same number of disjoint cycles of same length)
• Mar 21st 2011, 12:38 PM
poirot
Quote:

Originally Posted by Drexel28
A possibility is to decompose them into disjoint cycles then use the fact that two elements of $S_n$ are conjugate if and only if they have the same cycle structure (same number of disjoint cycles of same length)

so are (i) and (iii) conjugate?
• Mar 21st 2011, 12:49 PM
Drexel28
Quote:

Originally Posted by poirot
so are (i) and (iii) conjugate?

Hell if I know...I'm sorry I don't feel like wading through the cycle deomposition. Did you try it?
• Mar 21st 2011, 12:51 PM
poirot
Quote:

Originally Posted by Drexel28
Hell if I know...I'm sorry I don't feel like wading through the cycle deomposition. Did you try it?

those elelments are as a product of disjoint cycles so there is no further work.
• Mar 21st 2011, 12:53 PM
Drexel28
Quote:

Originally Posted by poirot
those elelments are as a product of disjoint cycles so there is no further work.

Hahaha! God bless you sir. Anyways, then I would ask you to inspect them and tell me if they have the same number of disjoint cycles (yes ;) ) and if the cycles have the same length (yes ;) )
• Mar 21st 2011, 01:08 PM
poirot
Quote:

Originally Posted by Drexel28
Hahaha! God bless you sir. Anyways, then I would ask you to inspect them and tell me if they have the same number of disjoint cycles (yes ;) ) and if the cycles have the same length (yes ;) )

haha lol. Yes I can count, I was just comfirming thats what you meant.