# Thread: conjugacy classes of A5

1. ## conjugacy classes of A5

"Prove that the 3-cycles in $A_5$ form a single conjugacy class"

Okay, I understand what a conjugacy class is (subset in which every element is conjugate to every other element in the conjugacy class), but I'm wondering exactly how I should go about proving this? Do I have to brute force my way, or is there some theorem or something I am simply forgetting about?

2. Originally Posted by jmoney90
"Prove that the 3-cycles in $A_5$ form a single conjugacy class"

Okay, I understand what a conjugacy class is (subset in which every element is conjugate to every other element in the conjugacy class), but I'm wondering exactly how I should go about proving this? Do I have to brute force my way, or is there some theorem or something I am simply forgetting about?

Theorem for you to find: a conjugacy class in S_n may remain a single conjugacy class or split in two conjugacy classes in A_n: it will split in two conjugacy classes iff there is no odd permutation with whom some representative of the conjugacy class commutes.

Tonio

3. Originally Posted by tonio
Theorem for you to find: a conjugacy class in S_n may remain a single conjugacy class or split in two conjugacy classes in A_n: it will split in two conjugacy classes iff there is no odd permutation with whom some representative of the conjugacy class commutes.

Tonio
Ahh, that makes things a lot easier. THank you!

4. Originally Posted by jmoney90
Ahh, that makes things a lot easier. THank you!

Any time.

Tonio

5. Originally Posted by tonio
Any time.

Tonio
Okay, just to make sure I have the right idea, I could take a 3-cycle such as (3 4 5) and then multiply it with (1 2) and it would commute since both cycles share no common numbers, and thus I prove that the 3-cycles make a conjugacy class in A_5 as per the theorem you provided, correct?

6. Originally Posted by jmoney90
Okay, just to make sure I have the right idea, I could take a 3-cycle such as (3 4 5) and then multiply it with (1 2) and it would commute since both cycles share no common numbers, and thus I prove that the 3-cycles make a conjugacy class in A_5 as per the theorem you provided, correct?

Yup...the interesting thing, though, is that you'd prove the theorem that allows you to deduce that, imo.

Tonio

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