Thread: Abelian subgroups of S_{n}

I am finding all abelian subgroups of a symmetric group 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: .

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 ?

Originally Posted by aliceinwonderland

I am finding all abelian subgroups of a symmetric group for n>=5.

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

to Landau that if is the maximum order of elements of then it's not surprising that there's no closed form for because is related to the partition function.

Originally Posted by NonCommAlg

i don't think this is basically possible to do because we even don't know all cyclic subgroups of no formula is known for the maximum order of elemens of

Any concrete example of an element of that has an order bigger than n? It is easy to find an element of order n in , but it is hard for me to find an element that has an order bigger than n.

Originally Posted by aliceinwonderland

Any concrete example of an element of that has an order bigger than n? It is easy to find an element of order n in , but it is hard for me to find an element that has an order bigger than n.

Sure, take and consider .