# Math Help - Sylow p-subgroup

1. ## Sylow p-subgroup

can someone explain what is meant by a Sylow p-subgroup

and hence outline the steps in the following question:

"Let G be a group. Prove that if G has only one Sylow p-subgroup H, then H is normal in G."

2. ## Re: Sylow p-subgroup

Originally Posted by ibnashraf
"Let G be a group. Prove that if G has only one Sylow p-subgroup H, then H is normal in G."
I think one of my problems here is that G was defined to be a group.
If G was defined to be a "finite group", then I could have said that by definition H is a subgroup of G.
However, there is no finite group mentioned in the question, so how exactly am I to proceed ?

3. ## Re: Sylow p-subgroup

Originally Posted by ibnashraf
I think one of my problems here is that G was defined to be a group.
If G was defined to be a "finite group", then I could have said that by definition H is a subgroup of G.
What? H is given as a subgroup so that is not in question.

However, there is no finite group mentioned in the question, so how exactly am I to proceed ?
The problem is to show that H is a normal subgroup. So what is the definition of "normal subgroup"?

4. ## Re: Sylow p-subgroup

Originally Posted by HallsofIvy
What? H is given as a subgroup so that is not in question.

The problem is to show that H is a normal subgroup. So what is the definition of "normal subgroup"?
ok well i guess since H is already a subgroup then i have to show that $g^{-1}hg \in H\Rightarrow$ H is normal in G ????

5. ## Re: Sylow p-subgroup

yes, you have to show that for any g in G, and h in H, that ghg-1 is in H.

equivalently (and easier to use for this problem), you need to show that gHg-1 is a subset of H, for any g in G.

a) prove that for any subgroup H of G, and any element g in G, the set gHg-1 = {ghg-1: h in H} is actually a subgroup of G.

b) show that H and gHg-1 have the same order.

conclude that if G has only ONE subgroup of a given order, that subgroup MUST be normal (sylow subgroup or not).

the definition of a sylow p-subgroup is, by the way:

a subgroup of order pk, where pk divides |G|, and pk+1 does not divide |G|.

for example, if G is a group of order 24, a sylow 2-subgroup of G is a subgroup of order 8, whereas a sylow 3-subgroup is a subgroup of order 3.