I am examining the conjugacy classes of S_5. I know that two elements in S_5 are conjugate if and only if they have the same structure when expressed as products of irreducible cycles. Thus, the conjugacy classes are [(1 2 3 4 5)], [(1 2 3 4)], [(1 2 3)], [(1 2 3)(4 5)], [(1 2)(3 4)], [(1 2)], [(1)]. Now, I want to find how many elements of S_5 are in each.(adsbygoogle = window.adsbygoogle || []).push({});

I need to find the number of cycles of length 2,3,4, and 5 in S_5. I know there are C(5,2)=10 cycles of length 2. I was thinking that to count the cycles of length 3, I start with a cycle of length 2, chose 1 of the 3 unused elements and insert it in one of two possible places to put it. (e.g. If I start with the cycle (1 2) and I choose to add 3, I can make (3 1 2) = (1 2 3) or (1 3 2) )

This should give C(10,1) * C(3,1) * 2 = 60 elements in [(1 2 3)]. However, since [(1 2 3)(4 5)] should have the same number of elements as [(1 2 3)] (since the 2-cycle is set once the 3-cycle is chosen), this would give more elements than are possible.

What am I forgetting to subtract off?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Counting cycles in S_5

**Physics Forums | Science Articles, Homework Help, Discussion**