Proving 1 p-Sylow Subgroup of G is Normal

In summary, a p-Sylow subgroup is a subgroup of a finite group with an order that is a power of a prime number p. It is the largest subgroup of this order within the group. To prove that a p-Sylow subgroup is normal, one must show that it is invariant under conjugation by elements of the group. The significance of proving this is that it simplifies the group's structure and allows for a better understanding of its properties. A group can have multiple p-Sylow subgroups, but they will all have the same order and be conjugate to each other. The Sylow theorems can be used to prove that a p-Sylow subgroup is normal by showing that the number of p
  • #1
dogma
35
0
Hello!

For the life of me, I can't seem to figure this out (vapor lock in the ol' brain):

Show that if G has only 1 p-Sylow subgroup, then it must be normal.

I know it something to do with showing it's a conjugate to itself (right coset = left coset?). I'm just not quite sure how to go about showing this.

Thanks for your time, help, and patience with me.

dogma
 
Physics news on Phys.org
  • #2
Let S be the sylow subgroup. gSg^{-1} is another sylow subgroup, and must be S as S is unique.
 
  • #3
thanks

Thank you...that makes sense.

dogma
 

1. What is a p-Sylow subgroup?

A p-Sylow subgroup is a subgroup of a finite group whose order is a power of a prime number p. It is the largest subgroup of the given order within the group.

2. How do you prove that a p-Sylow subgroup is normal?

To prove that a p-Sylow subgroup of G is normal, you need to show that it is invariant under conjugation by elements of G. This means that for any element g in G, gHg^-1 = H, where H is the p-Sylow subgroup. This can be done by showing that all the elements of H are conjugate to each other within G.

3. What is the significance of proving a p-Sylow subgroup is normal?

Proving that a p-Sylow subgroup is normal is important because it allows us to simplify the structure of the group. This allows us to focus on the normal subgroup and its cosets, making it easier to understand the group's properties and behavior.

4. Can a group have more than one p-Sylow subgroup?

Yes, a group can have multiple p-Sylow subgroups. However, all p-Sylow subgroups will have the same order and will be conjugate to each other, making them isomorphic. This is known as the Sylow theorems.

5. How do you use the Sylow theorems to prove that a p-Sylow subgroup is normal?

The Sylow theorems state that for any prime p, the number of p-Sylow subgroups in a finite group G is equal to 1 mod p, and it divides the order of G. If there is only one p-Sylow subgroup, then it must be normal. If there are multiple p-Sylow subgroups, they will all be conjugate to each other, meaning that they are all normal. Therefore, the Sylow theorems can be used to prove that a p-Sylow subgroup is normal.

Similar threads

MHB Show that the groups are abelian
    • Linear and Abstract Algebra
Replies
1
Views
788
I Is G simple?Is G simple? A different approach using Sylow theorems
    • Linear and Abstract Algebra
Replies
5
Views
1K
I Proving that a subgroup is normal
    • Linear and Abstract Algebra
Replies
8
Views
2K
MHB G contains a normal p-Sylow subgroup
    • Linear and Abstract Algebra
Replies
5
Views
2K
I Determining the number of conjugacy classes in a p-group
    • Linear and Abstract Algebra
Replies
6
Views
1K
MHB Normal subgroups (Sylow Theorem)
    • Linear and Abstract Algebra
Replies
2
Views
2K
MHB There is no proper subgroup
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Sylow subgroup of some factor group
    • Linear and Abstract Algebra
Replies
3
Views
1K
I First Sylow Theorem: Group of Order ##p^k## & Cyclic Groups
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Proof: Every Subgroup of Cyclic $H$ is Normal in $G$
    • Linear and Abstract Algebra
Replies
12
Views
3K

Hot Threads

Recent Insights

Back
Top