MHB Sylow p-Subgroups of Symmetric Group: Orders & Explanation

[ibnashraf]

Question:

What are the orders of the Sylow p-subgroups of the symmetric group

?
Give the possible orders of each Sylow p-subgroup of

.
(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

is the symmetric group of degree 6.
that is the symmetric group on {

}
and i think that the order is given by

.
where do i go from there?

 

[Swlabr1]

Question:

What are the orders of the Sylow p-subgroups of the symmetric group

?
Give the possible orders of each Sylow p-subgroup of

.
(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

is the symmetric group of degree 6.
that is the symmetric group on {

}
and i think that the order is given by

.
where do i go from there?

Well, what is a Sylow $p$-subgroup of a given group? What do you know about it? Well, it has order $p^n$ where $p^n$ divides the order of the group. So, what is the prime decomposition of $6!$? This will give you the list of possible primes $p$.

You now need to find an $n$ for each prime $p$. For this, you need to look at your notes on Sylow's theorems. One of the theorems will tell you what $n$ should be.

(Also, when you are using LaTeX you can put in curly brakets using \{ and \}. $\{1, 2, 3, 4, 5, 6\}$ looks much nicer than {$1, 2, 3, 4, 5, 6$} (you need to put a backslash before the curly brackets are curly brackets are part of LaTeX code - they "group" things together. For example, e^{\pi i} gives $e^{\pi i}$).)