Abstract Nonsense

Crushing one theorem at a time

Review of Group Theory: Alternate Proof of the Sylow Theorems (Pt. II)


Point of post: This is a continuation of this post.

We continue now with a proof what was previously called the third Sylow theorem. Namely, that the number of Sylow p-subgroups of a group is congruent to 1 modulo p. Namely:

 

Theorem: Let G be a finite group with |G|=p^\alpha m where p\nmid m. Then, if  \text{Syl}_p\left(G\right) denotes the set of all Sylow p-subgroups of G then \#\left(\text{Syl}_p\left(G\right)\right)\equiv 1\text{ mod }p.

Proof: Let P\in\text{Syl}_p\left(G\right) (we know that \text{Syl}_p\left(G\right) by our last theorem). Then, consider the P-action on \text{Syl}_p\left(G\right) given by

 

g\cdot S=gSg^{-1}

 

for g\in P and S\in\text{Syl}_P\left(G\right). Evidently \mathcal{O}_P=\{P\}, and we claim that this is the only orbit under this action with one element. To see this suppose that H\in\text{Syl}_p\left(G\right), \mathcal{O}_H=\{H\}, and H\ne P. Consider then that for each g\in P and h\in H we have that ghg^{-1}\in H by assumption. Thus the map

 

\displaystyle \varphi:P\longrightarrow \text{Aut}\left(H\right):g\mapsto i_g

 

where i_g is the variant of the inner automorphism given by i_g(h)=ghg^{-1} is well-defined. Thus, consider the group

 

H\rtimes_\varphi P

 

Note then that the map

 

\phi:H\rtimes_\varphi P\to G:(h,g)\mapsto hg

 

is a homomorphism. Indeed,

 

\displaystyle \begin{aligned}\phi\left((h,g)(h',g')\right) &= \phi\left(\left(h\varphi_g(h'),gg'\right)\right)\\ &= \phi\left(\left(hgh'g^{-1},gg'\right)\right)\\ &= hgh'g^{-1}gg'\\ &= hgh'g'\\ &= \phi\left(\left(h,g\right)\right)\phi\left(\left(h',g'\right)\right)\end{aligned}

 

Note though that \phi\left(\tilde{P}\right)=P and \phi\left(\tilde{H}\right)=H and since H\ne P we clearly must have that

 

\displaystyle \left|\phi\left(H\rtimes_\varphi P\right)\right|\geqslant \left|H\cap P\right|=|H|+|P|-\left|H\cap P\right|>p^\alpha

 

since H\cap P\ne H and H\cap P\ne P. Otherwise H\subseteq P or  P\subseteq H and since they are of the same cardinality it would follow that they are equal, contradicting our assumption. Note though that as a corollary of the First Isomorphism Theorem we have that

 

\displaystyle \left|\phi\left(H\rtimes_\varphi P\right)\right|\text{ divides }\left|H\rtimes_\varphi P\right|=p^{2\alpha}

 

and thus it follows that \left|\phi\left(H\rtimes_\varphi\right)\right|=p^k for some k\geqslant 0. But, recalling that \left|\phi\left(H\rtimes_\varphi\right)\right|>p^\alpha we may then conclude that \left|\phi\left(H\rtimes_\varphi P\right)\right|=p^\beta where \beta>\alpha. But, by Lagrange’s Theorem we may conclude that

 

p^\beta=\left|\phi\left(H\rtimes_\varphi P\right)\right|\text{ divides }|G|=p^\alpha m

 

but this clearly implies that p\mid m, which is a contradiction. It follows then that for every \mathcal{O}_H\in\text{Orb}\left(\text{Syl}_p\left(G\right)\right) with H\ne P we have that \#\left(\mathcal{O}_H\right)>1. But, by the Orbit-Stabilizer Theorem

 

\displaystyle \#\left(\mathcal{O}_H\right)=\left(P:G_H\right)=\frac{|P}{|G_H|}=\frac{p^\alpha}{|G_H|}

 

And so in particular we have that for every \mathcal{O}_H\in\text{Orb}\left(\text{Syl}_p\left(G\right)\right) we have that \#\left(\mathcal{O}_H\right)=p^r with r\in\mathbb{N}. But, from previous comment we know that for \mathcal{O}_H\in\text{Orb}\left(\text{Syl}_p\left(G\right)\right) with H\ne P we have that \#\left(\mathcal{O}_H\right)>1 and so in conjunction with the previous sentence we have that \#\left(\mathcal{O}_H\right)=p^k with k>0. It follows then from the Orbit Decomposition Theorem that

 

\displaystyle \begin{aligned}\#\left(\text{Syl}_p\left(G\right)\right) &=\sum_{\mathcal{O}_H\in\text{Orb}\left(\text{Syl}_p\left(G\right)\right)}\#\left(\mathcal{O}_H\right)\\ & \\ &=1+\sum_{\substack{\mathcal{O}_H\in\text{Orb}\left(\text{Syl}_p\left(G\right)\right)\\ H\ne P}}\#\left(\mathcal{O}_H\right)\\ & \\ &\equiv 1\text{ mod }p\end{aligned}

from where the conclusion follows. \blacksquare

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200

3.Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.

 

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January 12, 2011 - Posted by | Algebra, Group Theory | , ,

1 Comment »

  1. […] Point of post: This is a continuation of this post. […]

    Pingback by Review of Group Theory: Alternate Proof to the Sylow Theorems « Abstract Nonsense | January 12, 2011 | Reply


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