11-1100-Pgadey-Lect6: Difference between revisions

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==Theory of Transitive <math>G</math>-sets==
; Theorem
; Theorem
: Every <math>G</math>-set is a disjoint union of "transitive <math>G</math>-sets"
: Every <math>G</math>-set is a disjoint union of "transitive <math>G</math>-sets"
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; Stabilizer of a point
; Stabilizer of a point
: We write <math>Stab(x) = \{g \in G : gx = x\}</math> for the stabilizer subgroup of $x$.
: We write <math>Stab(x) = \{g \in G : gx = x\}</math> for the stabilizer subgroup of <math>x</math>.


''Proof'' We define an equivalence relation <math>x \sim y \iff \exists_{g \in G} gx = y</math>. This relation is reflexive since <math>x = ex</math> and thus <math>x \sim x</math>. This relation is symmetric since <math>y = gx</math> implies <math>g^{-1}y = x</math>. This relation is transitive, since if <math>x = gy</math> and <math>y = hz</math> then <math>x = ghz</math>. It follows that <math> X = \coprod_{i \in I} Gx_{i} </math> where <math>Gx_i</math> denote the orbit of a point <math>x_i</math>.
''Proof'' We define an equivalence relation <math>x \sim y \iff \exists_{g \in G} gx = y</math>. This relation is reflexive since <math>x = ex</math> and thus <math>x \sim x</math>. This relation is symmetric since <math>y = gx</math> implies <math>g^{-1}y = x</math>. This relation is transitive, since if <math>x = gy</math> and <math>y = hz</math> then <math>x = ghz</math>. It follows that <math> X = \coprod_{i \in I} Gx_{i} </math> where <math>Gx_i</math> denote the orbit of a point <math>x_i</math>.
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<math>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\}</math>
<math>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\}</math>


The last group <math>Q</math> is the famous ''unit quaternions'' -- They need a better description here.
The last group <math>Q</math> is the famous ''unit quaternion'' group (See [http://en.wikipedia.org/wiki/Quaternion_group], also [http://en.wikipedia.org/wiki/History_of_quaternions]).

; Theorem
; Theorem
: Any <math>p</math>-group has a non-trivial centre.
: Any <math>p</math>-group has a non-trivial centre.
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Observe that <math>|Gx_i|| = 1</math> iff <math>x_i \in Z(G)</math>. It follows that
Observe that <math>|Gx_i|| = 1</math> iff <math>x_i \in Z(G)</math>. It follows that
<math> |G| = |Z(G)| + \sum_{|Gx_i| > 1} \frac{|G|}{|Stab(x)|} </math>
<math> |G| = |Z(G)| + \sum_{|Gx_i| > 1} \frac{|G|}{|Stab(x)|} </math>
The formula above is called ''"the class formula"''. We have that <math>|G| / |Stab(x)| = p^k</math> for some <math> 1 < k</math> since <math>Stab(x)</math> is a subgroup. It follows that <math>|G| \equiv 0\ \mod\ p</math> and <math>\sum_{|Gx_i| > 1} \frac{|G|}{|Stab(x_i)|} \equiv 0\ \mod\ p</math>. It follows that <math>|Z(G)| \equiv 0\ \mod\ p</math>. Since <math>e \in Z(G)</math> we have <math>1 \leq |Z(G)|</math> and thus <math>p \leq |Z(G)|</math>.
The formula above is called ''"the class formula"''. We have that <math>|G| / |Stab(x)| = p^k</math> for some <math> 1 < k</math> since <math>Stab(x)</math> is a subgroup. It follows that <math>|G| \equiv 0\mod\ p</math> and <math>\sum_{|Gx_i| > 1} \frac{|G|}{|Stab(x_i)|} \equiv 0\mod\ p</math>. It follows that <math>|Z(G)| \equiv 0\mod\ p</math>. Since <math>e \in Z(G)</math> we have <math>1 \leq |Z(G)|</math> and thus <math>p \leq |Z(G)|</math>.


==Sylow==
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.
A prove a brief technical lemma, for fun, since we could deduce it from more high powered machinery which we don't have yet.
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; Sylow set
; Sylow set
: If <math>|G| = p^k m</math> for <math>m \not\equiv 0\ \mod\ p</math> then <math>Syl_p(G) = \{P \leq G : |P| = p^k</math>.
: If <math>|G| = p^k m</math> for <math>m \not\equiv 0\mod\ p</math> then <math>Syl_p(G) = \{P \leq G : |P| = p^k</math>.


; Sylow I
; Sylow I
: <math>Syl_p(G) \neq \emptyset</math>
: <math>Syl_p(G) \neq \emptyset</math>


We proceed by induction on the oder of $p$. Assume the claim holds for all groups of order less than $|G|$. <span style="color:green">[Dror: "Stare at the class equation.]</span> Since <math>|G| \equiv 0\ \mod\ p</math> we have either:
We proceed by induction on the order of <math>p</math>. Assume the claim holds for all groups of order less than <math>|G|</math>. <span style="color:green">[Dror: "Stare at the class equation."]</span> Since <math>|G| \equiv 0\mod\ p</math> we have either:
* <math>|G| \equiv 0\ \mod\ p</math> and <math>\sum |G|/|Stab(x_i)| \equiv 0\ \mod\ p</math>.
* <math>|G| \equiv 0\mod\ p</math> and <math>\sum |G|/|Stab(x_i)| \equiv 0\mod\ p</math>.
* <math>|G| \equiv 0\ \not\mod\ p</math> and <math>\sum |G|/|Stab(x_i)| \not\equiv 0\ \mod\ p</math>.
* <math>|G| \not\equiv 0\mod\ p</math> and <math>\sum |G|/|Stab(x_i)| \not\equiv 0\mod\ p</math>.


If <math>|Z(G)| \not\equiv 0\ \mod p</math> then there exists <math>x_i</math> such that <math>|G|/|Stab(x_i)| \not\equiv 0\ \mod\ p</math>. Thus <math>p^k</math> divides <math>|Stab(x_i)|</math>. We have that <math>|Stab(x_i)| < |G|</math> <span style="color:green">[Why happens here?]</span> We then have that <math>p^k \leq Stab(x_i) < |G|</math> and by induction there is <math>|P| = p^k</math> such that <math>P \leq Stab(x_i)</math>. It follows <math>P \leq Stab(x_i) \leq G</math>. We've obtained the Sylow <math>p</math>-subgroup.
If <math>|Z(G)| \not\equiv 0\ \mod p</math> then there exists <math>x_i</math> such that <math>|G|/|Stab(x_i)| \not\equiv 0\mod\ p</math>. Thus <math>p^k</math> divides <math>|Stab(x_i)|</math>. We have that <math>|Stab(x_i)| < |G|</math> We then have that <math>p^k \leq Stab(x_i) < |G|</math> and by induction there is <math>|P| = p^k</math> such that <math>P \leq Stab(x_i)</math>. It follows <math>P \leq Stab(x_i) \leq G</math>. We've obtained the Sylow <math>p</math>-subgroup.


WIf <math>|Z(G)| \equiv 0\ \mod p</math> then by Cauchy's Lemma, there is <math>x \in Z(G)</math> with <math>|<x>| = p</math>. Consider the group <math> G / <x> </math>. By the induction hypothesis there is <math> P' \leq G/<x> </math> where <math>|P'| = p^{k-1}</math>. Then, there is the canonical projection <math> \pi : G \rightarrow G/<x> </math>. By the fourth isomorphism theory <math> P = \pi^{-1}(P') \leq G </math> and <math> |\pi^{-1}(P')| = p(p^{k-1}) = p^k </math>.
WIf <math>|Z(G)| \equiv 0\ \mod p</math> then by Cauchy's Lemma, there is <math>x \in Z(G)</math> with <math>|<x>| = p</math>. Consider the group <math> G / <x> </math>. By the induction hypothesis there is <math> P' \leq G/<x> </math> where <math>|P'| = p^{k-1}</math>. Then, there is the canonical projection <math> \pi : G \rightarrow G/<x> </math>. By the fourth isomorphism theory <math> P = \pi^{-1}(P') \leq G </math> and <math> |\pi^{-1}(P')| = p(p^{k-1}) = p^k </math>.


; Sylow 2
; Sylow 2
: Every Sylow <math>p</math>-subgroup of <math>G> is conjugate. Moreover, every <math>p</math>-subgroup is contained in a Sylow <math>p</math>-subgroup.
: Every Sylow <math>p</math>-subgroup of <math>G</math> is conjugate. Moreover, every <math>p</math>-subgroup is contained in a Sylow <math>p</math>-subgroup.


; Sylow 3
; Sylow 3
: Let <math>n_p(G) = |Syl_p(G)|</math>. We have <math>n_p \equiv 0\ \mod\ |G|</math> and <math>n_p \equiv 1\ \mod\ p</math>.
: Let <math>n_p(G) = |Syl_p(G)|</math>. We have <math>n_p \equiv 0\mod\ |G|</math> and <math>n_p \equiv 1\mod\ p</math>.


; A Nearly Tautological Lemma
; A Nearly Tautological Lemma
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: If <math>x \in G</math> has <math>|<x>| = p^k</math> and <math>x \in N(P)</math> then <math>x \in P</math>.
: If <math>x \in G</math> has <math>|<x>| = p^k</math> and <math>x \in N(P)</math> then <math>x \in P</math>.


<span style="color:green">[Dror: "This lemma is nearly tautological but it is only nearly tautological once you understand that it is nearly tautological." Parker: "A tautology?"]
<span style="color:green">[Dror: "This lemma is nearly tautological but it is only nearly tautological once you understand that it is nearly tautological." Parker: "A tautology?"]</span>


We show the first statement. We have that <math>|P / P \cap H| = p^k</math> since <math>P</math> is a <math>p</math>-group. We then know that <math>PH / H \simeq P / P \cap H</math> by the second isomorphism theorem. It foolows that <math>|PH| = p^{k'}</math>. But since <math>P</math> is maximal, we have <math>P = PH</math> and thus <math>H \subseteq P</math>. The first statement implies the second by taking <math>H = <x> </math>.
We show the first statement. We have that <math>|P / P \cap H| = p^k</math> since <math>P</math> is a <math>p</math>-group. We then know that <math>PH / H \simeq P / P \cap H</math> by the second isomorphism theorem. It foolows that <math>|PH| = p^{k'}</math>. But since <math>P</math> is maximal, we have <math>P = PH</math> and thus <math>H \subseteq P</math>. The first statement implies the second by taking <math>H = <x> </math>.
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GROUPS OF ORDER 15
==Groups of Order 15==


If <math>|G| = 15</math> then <math>n_3 \equiv 0\ \mod\ 15</math> and <math>n_3 \equiv 1\ \mod\ 3</math>. These imply <math>n_3 = 1</math>. Moreover, <math>n_5 \equiv 0\ \mod\ 15</math> and <math>n_5 \equiv 1\ \mod\ 5</math>. These imply <math>n_5 = 1</math>. Thus we have <math>P_3</math> a normal <math>3</math>-subgroup. Moreover, we have <math>P_5</math> a normal <math>5</math>-subgroup. This tells us a lot about the group.
If <math>|G| = 15</math> then <math>n_3 \equiv 0\mod\ 15</math> and <math>n_3 \equiv 1\mod\ 3</math>. These imply <math>n_3 = 1</math>. Moreover, <math>n_5 \equiv 0\mod\ 15</math> and <math>n_5 \equiv 1\mod\ 5</math>. These imply <math>n_5 = 1</math>. Thus we have <math>P_3</math> a normal <math>3</math>-subgroup. Moreover, we have <math>P_5</math> a normal <math>5</math>-subgroup. This tells us a lot about the group.

Latest revision as of 15:22, 6 October 2011

Theory of Transitive 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} -sets

Theorem
Every 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} -set is a disjoint union of "transitive 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} -sets"
Theorem
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 X} is a transitive -set 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 x \in X} 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 X \simeq G/Stab(x)} where the isomorphism an isomorphism 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} -sets.
Transitive 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} -set
A 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} -set 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 transitive 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 \forall_{x,y \in X} \exists_{g \in G}\ st.\ gx = y} .
Stabilizer of a point
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 Stab(x) = \{g \in G : gx = x\}} for the stabilizer 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 x} .

Proof We define an equivalence relation . This relation is reflexive since and thus . This relation is symmetric since implies . This relation is transitive, since if and then . It follows that where denote the orbit of a point .

We then claim that is a transitive -set. [Dror: "[This fact] is too easy."]

We show that is isomorphic to as a -set.

We produce two morphism and .

To define there is only one thing we can do. We have and then we define . We check that this map is well defined. If then and hence . It follows that . Thus is well defined.

To define we take and define . We show that this map is well defined. If then and hence . It follows that and hence is well defined.

We need to check that and are mutually inverse and -set morphisms. We quickly check 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 \Phi} is a 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} -set morphism. 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 y = gx} 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 g_1 \in G} 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 g_1\Psi(y) = g_1(gStab(x)) = (g_1g)Stab(x)} . Similarly, 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 \Psi(g_1y) = g'Stab(x) = (g_1g)Stab(x)} . The last inequality follows since we can take any 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'} 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'y = g_1y} . Why not take -- since we know that works.

Theorem (Orbit-Stabilizer)
If 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 X = \coprod_{i \in I} Gx_i} 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 |X| = \sum_{i} \frac{|G|}{Stab(x_i)}} .

This is just a rewriting of the theorem above.

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} -Group
A 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} -group is a 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} 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 |G| = p^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} .

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 = (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 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 Q} is the famous unit quaternion group (See [1], also [2]).

Theorem
Any 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} -group has a non-trivial centre.

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 G} act on itself by conjugation. Decompose 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 = \coprod Gx_i} . 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 |G| = \sum_{|Gx_i| = 1} 1 + \sum_{|Gx_i| > 1} \frac{|G|}{|Stab(x)|} } Observe 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 |Gx_i|| = 1} iff 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 \in Z(G)} . 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 |G| = |Z(G)| + \sum_{|Gx_i| > 1} \frac{|G|}{|Stab(x)|} } The formula above is called "the class formula". 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 |G| / |Stab(x)| = p^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 1 < k} 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 Stab(x)} is a subgroup. 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 |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_{|Gx_i| > 1} \frac{|G|}{|Stab(x_i)|} \equiv 0\mod\ p} . 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 |Z(G)| \equiv 0\mod\ p} . 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 e \in Z(G)} 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 1 \leq |Z(G)|} 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 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 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 an abelian group 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 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|} , 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 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 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} . Assume the claim holds for all groups of order less than 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|} . [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|} 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 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 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 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 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 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_p(G) = |Syl_p(G)|} . 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 n_p \equiv 0\mod\ |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 n_p \equiv 1\mod\ p} .
A Nearly Tautological Lemma
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 \in Syl_p(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 H \leq N(P)} is a 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} -group, 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 H \leq 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 x \in G} has 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^k} 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 x \in N(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 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 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 \cap H| = p^k} 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 a 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} -group. We then know 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 PH / H \simeq P / P \cap H} by the second isomorphism theorem. It foolows 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 |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.

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