# Tag Archives: Sylow’s theorems

## otro lema estratégico

La siguiente lámina establece un lema vital para una demostración “más contemporánea” del Teorema de Sylow 1.

Filed under algebra, cucei math, group theory, what is math

## a poem, a strategic lemma

Lemma

$\cdot$  $p\in{\Bbb{P}}$

$\cdot$ $|G|=p^mr$ , $p\nmid r$

$\cdot$ $\forall H

$\cdot$ $\forall Q\in {\rm p\!-\!SS}_G$

$\Longrightarrow$

$\bullet$ $\exists g\in G$ such that $H\cap gQg^{-1}\in {\rm p\!-\!SS}_H$.

Proof:

By employing the Double Coset Counting Formula we have $|G|=\sum_a\frac{|H|\ |Q|}{|H\cap aQa^{-1}|}$, and since $p\nmid [G:Q]$ then $\exists b\in\{a\}$ such that $p\nmid [H:H\cap bQb^{-1}]$.

But $H\cap bQb^{-1} so $|H\cap bQb^{-1}|=p^l\ ,\ \exists l\in{\Bbb{N}}$, hence, having $p\nmid\frac{|H|}{p^l}$, this implies that $|H|=p^l\alpha$, where $p\nmid\alpha\ ,\ \exists \alpha\in{\Bbb{N}}$.

Then, for $H\cap bQb^{-1} with $|H\cap bQb^{-1}|=p^l$ we deduce it is ${\rm p\!-\!SS}_H$

$\Box$

double coset counting formula

Filed under algebra, group theory, math, word algebra

## double coset counting formula

the double coset counting formula is a relation inter double cosets $HaK$, where $a\in G$ and $H,K$ subgroups in $G$. This is:

$\#(HaK)=\frac{|H||K|}{|H\cap aKa^{-1}|}$

and

$\#(G/K)=\sum_a[H;H\cap aKa^{-1}]$

The proof is easy.

One is to be bounded to the study of the natural map $H\times K\stackrel{\phi_a}\to HaK$. And it uses the second abstraction lemma.

The formula allows you to see the kinds of subgroups of arbitrary $H$ versus $K$ a $p-SS$ of $G$, $p-SS$ for the set of the $p$– Sylow subgroups.

Or, you can see that through the action $H\times G/K\to G/K$ via $h\cdot aK=haK$ you can get:

• ${\rm Orb}_H(aK)=\{haK\}$ which comply the equi-partition
• $HaK=aK\sqcup haK\sqcup...\sqcup h_taK$, so $\#(HaK)=m|K|$, for some $m\in \mathbb{N}$
• ${\rm St}_H(aK)=H\cap aKa^{-1}$

then you can deduce:

$|G|=\sum_a\frac{|H||K|}{|H\cap aKa^{-1}|}$

Now, let us use those ideas to prove the next statement:

Let $G$ be a finite group, with cardinal $|G|=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}$, where each $q_i$ are primes with $q_1 and $n_i$ positive integers.

Let $H$ be a subgroup of $|G|$ of index $[G:H]=q_1$.

Then, $H$ is normal.

Proof:

By employing $K=H$ in the double coset partition, one get the decomposition:

$G=HeH\sqcup Ha_1H\sqcup...\sqcup Ha_tH$

So by the double coset counting formula you arrive to:

$|G/H|=1+[H:H\cap a_1Ha_1^{-1}]+\cdots+[H:H\cap a_tHa_t^{-1}]$

i.e.

$q_1=1+\frac{|H|}{|H\cap a_1Ha_1^{-1}|}+\cdots+\frac{|H|}{|H\cap a_tHa_t^{-1}|}$

From this, we get $\frac{|H|}{|H\cap a_iHa_i^{-1}|}.

But $|G|=q_1|H|$ as well $|H|=|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}]$ so

$|G|=q_1|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}]$, i.e.

$[H:H\cap a_iHa_i^{-1}]$ divides $|G|$

Then $[H:H\cap a_iHa_i^{-1}]=1$. So $|H|=|H\cap a_iHa_i^{-1}|$ for each $a_i$.

This implies $H=H\cap a_iHa_i^{-1}$ and so $H=a_iHa_i^{-1}$ for all the posible $a_i$, hence, $H$ is normal.

QED.

## fourth Sylow theorem

We are going to reconstruct the salient of a part of the classical theorem:

Each $H$ p-S of $G$ is contained into a $Q$ p-SS of $G$

Here the reference frame:

1. $\#$p-SS divides $|G|$:

The set p-SS must be considered as an orbit of the action $G\times$ p-SS $\to$ p-SS via $g\cdot Q=gQg^{-1}$.  Since from sylow II we know that each two are conjugated the there is only one orbit

p-SS $={\rm Orb}_G(Q)\leftrightarrow G/{\rm St}_G(Q)$

giving us a trivial  orbital partition. Then $\#$p-SS=$|{\rm Orb}_G(Q)|$, i.e.

$\#$p-SS=$=[G:{\rm St}_G(Q)]=[G:N_G(Q)]$,

because for the isotropy group is ${\rm St}_G(Q)=\{\!x\!\in\!G: xQx^{-1}\!=\!Q\}=N_G(Q)\!\}$.

Then $|G|=\#{\rm p\!-\!SS}|N(Q)|$.

2. Observe also that $p\not|$#p-SS:

since $|G|=p^mr$ and $|G|=|N_G(Q)|[G:N_G(Q)]$,  as  far as $|N_G(Q)|=|Q|[N_G(Q):Q]$

so

$p^mr=|G|$

$=|Q|[G:N_G(Q)][N_G(Q):Q]$

$=p^m[N_G(Q):Q]\cdot\#$p-SS

which implies that $r=[N_G(Q):Q]\cdot\#$p-SS, then $p\!\!\not|\#$p-SS.

Proof of Theorem:

Considering the conjugation sub-action

$H\times$ p-SS$\to$ p-SS,

we get a orbit decomposition p-SS = ${\rm Orb}_H(Q_1)\sqcup\cdots\sqcup{\rm Orb}_H(Q_t)$ with the corresponding class equation

#p-SS = $[H:N_H(Q_1)]+\cdots+[H:N_H(Q_t)]$,

but asumming that $|H|=p^{\alpha}$ then

#p-SS = $\frac{p^{\alpha}}{|N_H(Q_1)|}+\cdots+\frac{p^{\alpha}}{|N_H(Q_t)|}$.

Now since $p\!\!\not|$#p-SS then there is some $Q_i$ for which $|N_H(Q_i)|=|H|$, so

$H=N_H(Q_i)$.

With that, it is easy these: $HQ_i and $Q_i$ is normal in $HQ_i$.

Also $\pi:H\to HQ_i/Q_i$ given by $x\mapsto\pi(x)=xQ_i$  is an epimorphism, so by the fundamental theorem of group-morphisms we have

$\frac{HQ_i}{Q_i}\cong\frac{H}{H\cap Q_i}$,

Observing that $H,\ Q_i and $|HQ_i|=[H:H\cap Q_i]|Q_i|$ then

$|HQ_i|=\frac{|H||Q_i|}{|H\cap Q_i|}=p^{m+\alpha-\beta}$,

where $|H\cap Q_i|=p^{\beta}$,

but $m$ is maximal, then $\alpha-\beta=0$. Hence $|H|=|H\cap Q_i|$ and

$H=H\cap Q_i.

$\Box$

Confer Milne Chapter 5