# Thread: Symmetric group

1. ## Symmetric group

Question:

What are the orders of the Sylow p-subgroups of the symmetric group $\displaystyle S_6$?
Give the possible orders of each Sylow p-subgroup of $\displaystyle S_6$.
(N.B. If there are many possible orders, then give at least four).

Can anyone help me to understand what is meant by the above question please?

So far i understand that $\displaystyle S_6$ is the symmetric group of degree 6.
that is the symmetric group on {$\displaystyle {1,2,3,4,5,6}$ }
and i think that the order is given by $\displaystyle 6!$.
where do i go from there?

2. ## Re: Symmetric group

the primes dividing 6! are 2,3, and 5 (7 is too big).

the prime factorization of 6! = 720 is: (24)(32)(5).

the sylow 2-subgroups are all of order 16.

the sylow 3-subgroups are all of order 9.

the sylow 5-subgroups are all of order 5.

3. ## Re: Symmetric group

Thanks alot Deveno .... i finally understand how to go about finding Sylow p-subgroups now.
my lecture notes in class wasn't to clear on this ....