Thread: 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?

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 are conjugate if and only if they have the same cycle structure (same number of disjoint cycles of same length)

Originally Posted by Drexel28

A possibility is to decompose them into disjoint cycles then use the fact that two elements of 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?

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?

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.

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 )

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.