MHB Why Can't a Proper Normal Subgroup Contain a Sylow Normalizer in a Group?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Subgroup
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

Let $P$ be a $p$-Sylow subgroup in $G$ and $N=N_G(P)$.

I want to show that there is no proper normal subgroup $H$ of $G$ that contains $N$.
We suppose that there is a proper normal subgroup $H$ of $G$ that contains $N$, $$N\leq H<G$$

Then $[G:N]=[G:H][H:N]$, with $[G:H]>1$.

How can we find a contradicion? (Wondering)

Do we use the definition of a normal subgroup? (Wondering)
 
Physics news on Phys.org
Do we maybe use Frattini Argument? (Wondering)

By Frattini Argument we have that $G=HN_G(P)$.

Since $H$ is a normal subgroup of $H$ and since $N_G(P)\leq H$, we have that $HN_G(P)\subseteq H \Rightarrow G\subseteq H$.

We have that $H\subseteq G$.

So, it holds that $G=H$. This is a contradiction, since $H$ is a proper subgroup of $G$. Is this correct? (Wondering)
 
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...

Similar threads

Back
Top