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.

truccu

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a poem, a strategic lemma


Lemma

\cdot  p\in{\Bbb{P}}

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

\cdot \forall H<G

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

\Box

DCCF01

double coset counting formula

 

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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<q_2<...<q_t 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}|}<q_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.

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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<G 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<HQ_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<Q_i.

\Box

Confer Milne Chapter 5

数学

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