# Abelian subgroups of S_{n}

• Jan 6th 2009, 06:03 PM
aliceinwonderland
Abelian subgroups of S_{n}
I am finding all abelian subgroups of a symmetric group $\displaystyle S_{n}$ for n>=5.

What I have found so far is
1. {e} : trivial group
2. a cyclic group of order n
3. a quotient group of order 2: $\displaystyle S_{n}/A_{n}$.

I am wondering if below 4 is correct.
4. "a cyclic group of order k which is a divisor of n"

Any more subgroup exists for $\displaystyle S_{n}$?
• Jan 6th 2009, 07:53 PM
NonCommAlg
Quote:

Originally Posted by aliceinwonderland

I am finding all abelian subgroups of a symmetric group $\displaystyle S_{n}$ for n>=5.

i don't think this is basically possible to do because we even don't know all cyclic subgroups of $\displaystyle S_n.$ no formula is known for the maximum order of elemens of $\displaystyle S_n.$ however, there's a result due

to Landau that if $\displaystyle f(n)$ is the maximum order of elements of $\displaystyle S_n,$ then $\displaystyle \lim_{n\to\infty} \frac{\ln f(n)}{\sqrt{n \ln n}} = 1.$ it's not surprising that there's no closed form for $\displaystyle f(n),$ because $\displaystyle f(n)$ is related to the partition function.
• Jan 6th 2009, 08:56 PM
aliceinwonderland
Quote:

Originally Posted by NonCommAlg
i don't think this is basically possible to do because we even don't know all cyclic subgroups of $\displaystyle S_n.$ no formula is known for the maximum order of elemens of $\displaystyle S_n.$

Any concrete example of an element of $\displaystyle S_n$ that has an order bigger than n? It is easy to find an element of order n in $\displaystyle S_n$, but it is hard for me to find an element that has an order bigger than n.
• Jan 6th 2009, 09:01 PM
ThePerfectHacker
Quote:

Originally Posted by aliceinwonderland
Any concrete example of an element of $\displaystyle S_n$ that has an order bigger than n? It is easy to find an element of order n in $\displaystyle S_n$, but it is hard for me to find an element that has an order bigger than n.

Sure, take $\displaystyle S_7$ and consider $\displaystyle (1234)(567)$. (Wink)