Number of conjugated to (12)(34) in $S_6$ of rank 6

Question

I need to find the number of conjugated to the permutation (12)(34) in the symmetric group $S_6$ of rank 6

My answer is 6! = 720

Is this correct?

I concluded that (12)(34)=(12)(34)(5)(6) and the number of combinations for $S_6$ is 6! as they need to be the same partition type

Edit:

It seems to be $6! / (2*2*1*1) = 180$

$6!$ would mean that every permutation is a conjugate of this one. Do you know the criterion with cycle decomposition? — Berci, Jan 03 '21 at 10:17

score 2 · Accepted Answer · answered Jan 03 '21 at 10:47

2

Hint: An element of $S_6$ is conjugated with $(12)(34)$ if it has the same disjoint cycle type, that is you have to count the amount of elements that look like $(a b)(cd)$ such that $(ab)$ and $(cd)$ are disjoint transpositions.

answered Jan 03 '21 at 10:47

J. De Ro

21,438

Ok, so it is 105? – Daniel Jan 03 '21 at 11:09
Nope. It is $45$. I will help you to count this. How many choices do you have for $(a b)$ to 'fill in' $a$ and $b$? – J. De Ro Jan 03 '21 at 11:11

score 2 · Answer 2 · answered Jan 03 '21 at 11:03

2

The number of permutations with cycle type $(2,2,1,1)$ is ${6\choose2}\cdot{4\choose2}/2=45$. (I divided by two because there is double counting.)

answered Jan 03 '21 at 11:03

Number of conjugated to (12)(34) in $S_6$ of rank 6

2 Answers2