11-1100-Pgadey-Lect6

Theory of Transitive $G$-sets

Theorem

Every $G$-set is a disjoint union of "transitive $G$-sets"

Theorem

Theorem

If $X$ is a transitive $G$-set and $x\in X$ then $X\simeq G/Stab(x)$ where the isomorphism an isomorphism of $G$-sets.

Transitive $G$-set

A $G$-set $X$ is transitive is $\forall _{x,y\in X}\exists _{g\in G}\ st.\ gx=y$.

Stabilizer of a point

We write $Stab(x)=\{g\in G:gx=x\}$ for the stabilizer subgroup of $x$.

Proof We define an equivalence relation $x\sim y\iff \exists _{g\in G}gx=y$. This relation is reflexive since $x=ex$ and thus $x\sim x$. This relation is symmetric since $y=gx$ implies $g^{-1}y=x$. This relation is transitive, since if $x=gy$ and $y=hz$ then $x=ghz$. It follows that $X=\coprod _{i\in I}Gx_{i}$ where $Gx_{i}$ denote the orbit of a point $x_{i}$.

We then claim that $Gx_{i}$ is a transitive $G$-set. [Dror: "[This fact] is too easy."]

We show that $Gx$ is isomorphic to $G/Stab(x)$ as a $G$-set.

We produce two morphism $\Psi :Gx\rightarrow G/Stab(x)$ and $\Phi :G/Stab(x)\rightarrow Gx$.

To define $\Psi$ there is only one thing we can do. We have $y\in Gx\Rightarrow y=gx$ and then we define $\Psi (y)=gStab(x)$. We check that this map is well defined. If $y=gx=g'x$ then $g^{-1}g'x=x$ and hence $g^{-1}g\in Stab(x)$. It follows that $gStab(x)=g'Stab(x)$. Thus $\Psi$ is well defined.

To define $\Phi$ we take $gStab(x)\in G/Stab(x)$ and define $\Phi (gStab(x))=gx$. We show that this map is well defined. If $gStab(x)=g'Stab(x)$ then $g^{-1}g'\in Stab(x)$ and hence $g^{-1}g'x=x$. It follows that $gx=g'x$ and hence $\Phi$ is well defined.

We need to check that $\Psi$ and $\Phi$ are mutually inverse and $G$-set morphisms. We quickly check that $\Phi$ is a $G$-set morphism. If $y=gx$ and $g_{1}\in G$ then $g_{1}\Psi (y)=g_{1}(gStab(x))=(g_{1}g)Stab(x)$. Similarly, $\Psi (g_{1}y)=g'Stab(x)=(g_{1}g)Stab(x)$. The last inequality follows since we can take any $g'$ such that $g'y=g_{1}y$. Why not take $g'=g_{1}g$ -- since we know that works.

Theorem (Orbit-Stabilizer)

If $|X|<\infty$ and $X=\coprod _{i\in I}Gx_{i}$ then $|X|=\sum _{i}{\frac {|G|}{Stab(x_{i})}}$.

This is just a rewriting of the theorem above.

$p$-Group

A $p$-group is a group $G$ with $|G|=p^{k}$ for some $k$.

$G=(Z/2Z)^{3},(Z/2Z)\times (Z/4Z),Z/8Z,D_{8},Q=\{\pm 1,\pm i,\pm j,\pm k:i^{2}=j^{2}=k^{2}=-1,ij=k\}$

The last group $Q$ is the famous unit quaternions -- They need a better description here.

Theorem

Any $p$-group has a non-trivial centre.

Let $G$ act on itself by conjugation. Decompose $G=\coprod Gx_{i}$. Then, $|G|=\sum _{|Gx_{i}|=1}1+\sum _{|Gx_{i}|>1}{\frac {|G|}{|Stab(x)|}}$ Observe that $|Gx_{i}||=1$ iff $x_{i}\in Z(G)$. It follows that $|G|=|Z(G)|+\sum _{|Gx_{i}|>1}{\frac {|G|}{|Stab(x)|}}$ The formula above is called "the class formula". We have that $|G|/|Stab(x)|=p^{k}$ for some $1<k$ since $Stab(x)$ is a subgroup. It follows that $|G|\equiv 0\mod \ p$ and $\sum _{|Gx_{i}|>1}{\frac {|G|}{|Stab(x_{i})|}}\equiv 0\mod \ p$. It follows that $|Z(G)|\equiv 0\mod \ p$. Since $e\in Z(G)$ we have $1\leq |Z(G)|$ and thus $p\leq |Z(G)|$.

Sylow

A prove a brief technical lemma, for fun, since we could deduce it from more high powered machinery which we don't have yet.

Cauchy's Lemma

If $A$ is an abelian group and $p$ divides $|A|$, then there is an element of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

Proof. Pick Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in A} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} divides the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} then we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{np} = e} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . It follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x^n)^p = e} . We then have that the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} does not divide the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , then consider Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A / <x> } . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is abelian, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <x> } is a normal subgroup. We have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A/<x>|} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A / <x>| < |A|} . We then induct. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y<x> } have order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , that is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y<x>)^p = <x> } . We then have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^p = x^k } for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k } . We write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |<y>| = np + r } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq p < r } . We then have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = y^{|<y>|} = y^{np + r} = x^{nkp}y^r \Rightarrow y^r \in <x> } . It follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y<x>)^r = <x> } contradicting the assumption that the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y<x> } is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } .

Sylow set

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G| = p^k m} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \not\equiv 0\mod\ p} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Syl_p(G) = \{P \leq G : |P| = p^k} .

Sylow I

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Syl_p(G) \neq \emptyset}

We proceed by induction on the oder of $p$. Assume the claim holds for all groups of order less than $|G|$. [Dror: "Stare at the class equation."] Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G| \equiv 0\mod\ p} we have either:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G| \equiv 0\mod\ p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum |G|/|Stab(x_i)| \equiv 0\mod\ p} .
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G| \not\equiv 0\mod\ p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum |G|/|Stab(x_i)| \not\equiv 0\mod\ p} .

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |Z(G)| \not\equiv 0\ \mod p} then there exists Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G|/|Stab(x_i)| \not\equiv 0\mod\ p} . Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^k} divides Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |Stab(x_i)|} . We have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |Stab(x_i)| < |G|} [Why happens here?] We then have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^k \leq Stab(x_i) < |G|} and by induction there is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P| = p^k} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \leq Stab(x_i)} . It follows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \leq Stab(x_i) \leq G} . We've obtained the Sylow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -subgroup.

WIf Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |Z(G)| \equiv 0\ \mod p} then by Cauchy's Lemma, there is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in Z(G)} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |<x>| = p} . Consider the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G / <x> } . By the induction hypothesis there is $P'\leq G/<x>$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P'| = p^{k-1}} . Then, there is the canonical projection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi : G \rightarrow G/<x> } . By the fourth isomorphism theory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \pi^{-1}(P') \leq G } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\pi^{-1}(P')| = p(p^{k-1}) = p^k } .

Sylow 2

Every Sylow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G> is conjugate. Moreover, every <math>p} -subgroup is contained in a Sylow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -subgroup.

Sylow 3

Let $n_{p}(G)=|Syl_{p}(G)|$. We have $n_{p}\equiv 0\mod \ |G|$ and $n_{p}\equiv 1\mod \ p$.

A Nearly Tautological Lemma

If $P\in Syl_{p}(G)$ and $H\leq N(P)$ is a $p$-group, then $H\leq P$.

If $x\in G$ has $|<x>|=p^{k}$ and $x\in N(P)$ then $x\in P$.

[Dror: "This lemma is nearly tautological but it is only nearly tautological once you understand that it is nearly tautological." Parker: "A tautology?"]

We show the first statement. We have that $|P/P\cap H|=p^{k}$ since $P$ is a $p$-group. We then know that $PH/H\simeq P/P\cap H$ by the second isomorphism theorem. It foolows that $|PH|=p^{k'}$. But since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is maximal, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = PH} and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \subseteq P} . The first statement implies the second by taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = <x> } .

Groups of Order 15

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |G| = 15} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_3 \equiv 0\mod\ 15} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_3 \equiv 1\mod\ 3} . These imply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_3 = 1} . Moreover, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_5 \equiv 0\mod\ 15} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_5 \equiv 1\mod\ 5} . These imply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_5 = 1} . Thus we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_3} a normal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} -subgroup. Moreover, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_5} a normal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5} -subgroup. This tells us a lot about the group.