# Sylow p-subgroups

#### ibnashraf

##### New member
Question:

What are the orders of the Sylow p-subgroups of the symmetric group
$S_6$
?
Give the possible orders of each Sylow p-subgroup of
$S_6$
.
(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
$S_6$
is the symmetric group of degree 6.
that is the symmetric group on {
${1,2,3,4,5,6}$
}
and i think that the order is given by
$6!$
.
where do i go from there?

#### Swlabr

##### New member
Question:

What are the orders of the Sylow p-subgroups of the symmetric group
$S_6$
?
Give the possible orders of each Sylow p-subgroup of
$S_6$
.
(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
$S_6$
is the symmetric group of degree 6.
that is the symmetric group on {
${1,2,3,4,5,6}$
}
and i think that the order is given by
$6!$
.
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 $e^{\pi i}$).)