MHB Normal subgroups (Sylow Theorem)

  • Thread starter Thread starter mathgirl1
  • Start date Start date
  • Tags Tags
    Normal Theorem
mathgirl1
Messages
21
Reaction score
0
Let G be a group of order pm where p is a prime and p > m. Suppose H is a subgroup of order p. Show that H is normal in G.

There is a very similar problem

Let |G| = p^nm where p is a prime and n \ge 1, p > m. Show that the Sylow p-subgroup of G is normal in G.

Proof:
Let n_p be the number of Sylow p-subgroups of G. By the 3-d Sylow Subgroup Theorem we know that n_p \mid m and that n_p \equiv 1 (mod p). Since m < p and n_p \mid m, it follows that 1 \le n_p \le m &lt; p. Since we also know that n_p \equiv 1 (mod p), it follows that n_p=1. Let P be a Sylow p-subgroup of G. Since for every g \in G, g^{-1}Pg is also a Sylow p-subgroup of G and since n_p=1, it follows that for every g \in G that g^{-1}Pg=P. Hence P is normal in G, as claimed.

So this is the same problem that I need to solve with n=1. Is there a way to solve this without Sylow Theorems or Sylow subgroups? We barely covered that so not sure we can use it to prove this problem.

Also, why is g^{-1}Pg also a Sylow p-subgroup of G since P is?

Any help is appreciated. I would like to solve this problem without Sylow theorems but not sure where to start cause I am already stuck in this thinking.

Thanks!
 
Physics news on Phys.org
You have shown above that there is exactly one subgroup of order $p$. Let us call it $P$. Furthermore, this is a Sylow $p$-subgroup because $p$ is the largest power of $p$ which divides $|G|=pm$. Thus, by the third theorem all Sylow $p$-subgroups are conjugate. Consider $gPg^{-1}$. Show that this subgroup has order $p$, and so, it is a Sylow $p$-subgroup. But since there is only one such subgroup it follows that $gPg^{-1} = P$. Since this is true for all $p$ we conclude that $P$ is invariant under conjugation, so it is a normal subgroup.
 
Hi,
You asked for a different proof that doesn't use the Sylow theorems. Here is one. The idea of representing (finding a homomorphism) a group G on the cosets of a subgroup is worth remembering.

nx43sw.png
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top