Sylow p-subgroup

• Jun 17th 2012, 02:38 PM
ibnashraf
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."
• Jun 17th 2012, 03:12 PM
ibnashraf
Re: Sylow p-subgroup
Quote:

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 ?
• Jun 17th 2012, 05:31 PM
HallsofIvy
Re: Sylow p-subgroup
Quote:

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.

Quote:

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"?
• Jun 17th 2012, 05:41 PM
ibnashraf
Re: Sylow p-subgroup
Quote:

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 $\displaystyle g^{-1}hg \in H\Rightarrow$ H is normal in G ????
• Jun 18th 2012, 05:49 AM
Deveno
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.