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A: Proof that $\langle y\rangle$ is a normal subgroup of $G$ with order $q$ and $|G|=pq$ with $p$,$q$ prime

Maurizio MoreschiThis is an instance of the following more general result. If $G$ is a finite group of order $n$ and $p$ is the smallest prime divisor of $n$, then any subgroup $H$ of $G$ of index $p$ is normal in $G$. Proof. Consider the left action of $G$ on the set of left cosets of $H$ in $G$, and let $\ph...

The Coding Wombat
The Coding Wombat
And in my case the subgroup $\langle y \rangle$ of $G$ is of index $p$ because $\frac{|G|}{|\langle y \rangle |}=p$, correct?
Maurizio Moreschi
Maurizio Moreschi
Yes, and because $p<q$.
The Coding Wombat
The Coding Wombat
What does $S_p$ mean?
Maurizio Moreschi
Maurizio Moreschi
Oh, sorry. I denote $S_p$ the group of permutations of a set with $p$ elements (in this case the set of left cosets of $H$ in $G$).
The Coding Wombat
The Coding Wombat
So $S_p=G / H$? (the quotient group)
Maurizio Moreschi
Maurizio Moreschi
13:10
Suppose $\Omega=\{g_1 H,\dots, g_p H\}$ is the set of left closets of $H$ in $G$ (i.e. the orbits of the action on $G$ on $H$ by left multiplication). Then $G$ acts on $\Omega$ by $g\cdot g_i H=(gg_i)H$. This means that you have a homomorphism of group $\phi:G\to Sym(\Omega)=:S_p$.
The Coding Wombat
The Coding Wombat
I hope we can continue talking here.
Your last comment made it probably more clear, but it's very technical for me
Maurizio Moreschi
Maurizio Moreschi
Yes, don't worry. Which part is not clear?
The Coding Wombat
The Coding Wombat
That omega group is the same as G/H right?
Maurizio Moreschi
Maurizio Moreschi
The point is, a priori you don't know it is a group
That is equivalent to the fact that $H$ is normal in $G$
Sym(\Omega) is the group of permutation of the elements of $\Omega$
The Coding Wombat
The Coding Wombat
Okay, so Omega is the same as the set of G/H
Maurizio Moreschi
Maurizio Moreschi
13:16
The group law being given by composition
The Coding Wombat
The Coding Wombat
So what elements does Sym(\Omega) contain?
Maurizio Moreschi
Maurizio Moreschi
A permutation of $\Omega$ is a function $\Omega\to \Omega$
The Coding Wombat
The Coding Wombat
So Sym(Omega) is a collection of sets with each set the elements g1H, g2H, ... gpH in a different order?
Maurizio Moreschi
Maurizio Moreschi
Yes
The Coding Wombat
The Coding Wombat
So every element of G is mapped to a set in S_p by the homomorphism
But later on you say G/K is a subgroup of Sp, but I thought Sp was not yet proven to be a set?
And K is the kernel of the homomorphism, so doesn't that mean that \phi (K) = {identy element of Sp} and that as only element?
Maurizio Moreschi
Maurizio Moreschi
13:50
sorry, that was a typo, I meant $\phi(G)$
The Coding Wombat
The Coding Wombat
That's okay, I'm glad I at least have some understanding of this to catch that :). Group theory is quite notation heavy and overwhelming
The Coding Wombat
The Coding Wombat
14:05
Okay, so what does the kernel of this homomorphism mean? What is the identity of the collection of permutations of omega?
The Coding Wombat
The Coding Wombat
14:20
And proving that <x> is also normal? q is not the smallest prime, so I guess I can't use the same proof

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