Sylow Theorems Sylow Theorems Andrew Clarey Looking at the Structure of Arbitrary Groups Definitions/ Theorems Groups, Subgroups Lagrange’s, Normality Class Equation, Andrew Clarey Cauchy’s Theorem First Sylow Theorem Occidental College Theorem Examples Mentor : Professor Nalsey Tinberg Proof Additional Proofs Second Sylow December 3, 2015 Theorem Third Sylow Theorem All material comes from Saracino, Abstract Algebra unless Results otherwise stated. Cyclic subgroups Simple Groups Additional Examples 1 / 26 References
Overview Definitions/ Theorems 1 Sylow Theorems Groups, Subgroups Andrew Clarey Lagrange’s, Normality Class Equation, Cauchy’s Theorem Definitions/ Theorems First Sylow Theorem 2 Groups, Subgroups Lagrange’s, Theorem Normality Class Equation, Examples Cauchy’s Theorem Proof First Sylow Theorem Additional Proofs Theorem Examples Second Sylow Theorem 3 Proof Additional Proofs Third Sylow Theorem 4 Second Sylow Results 5 Theorem Cyclic subgroups Third Sylow Theorem Simple Groups Results Additional Examples Cyclic subgroups Simple Groups References 6 Additional Examples 2 / 26 References
Definitions/Theorems Sylow Theorems A set G is called a group [denoted ( G , ∗ )] if: Andrew Clarey i) G has a binary operator ∗ . We write a ∗ b as ab . Definitions/ Theorems ii) ∗ is associative Groups, Subgroups iii) there is an element e ∈ G such that Lagrange’s, Normality x ∗ e = e ∗ x = x , ∀ x ∈ G Class Equation, Cauchy’s Theorem iv) for each x ∈ G , ∃ y ∈ G such that x ∗ y = y ∗ x = e . We First Sylow write y = x − 1 . Theorem Theorem A group G is called cyclic if ∃ x ∈ G such that Examples Proof G = { x n | n ∈ Z } = � x � . Then x is called a generator . Additional Proofs Second Sylow Example cyclic groups are Z , Z n . Theorem The order of a group G , denoted | G | , is the number of Third Sylow Theorem elements in the group. Results Cyclic subgroups Simple Groups Additional Examples 3 / 26 References
Definitions/Theorems Sylow A subset H of a group ( G , ∗ ) is called a subgroup of G if Theorems all h ∈ H form a group under ∗ . Andrew Clarey Definitions/ Theorem : Let H be a nonempty subset of a group G . Theorems Groups, Subgroups Then H is a subgroup iff: Lagrange’s, Normality i) ∀ a , b ∈ H , ab ∈ H Class Equation, Cauchy’s Theorem ii) ∀ a ∈ H , a − 1 ∈ H First Sylow Theorem We write H ≤ G . Theorem If H � G , then a Left/Right coset of H in G is a subset Examples Proof of the form aH / Ha where a ∈ G and Additional Proofs Second Sylow aH / Ha = { ah / ha | h ∈ H } . Theorem Two elements x , y ∈ G are conjugate if ∃ g ∈ G such that Third Sylow Theorem y = g − 1 xg . If H � G , then gHg − 1 � G is a conjugate subgroup of Results Cyclic subgroups G , ∀ g ∈ G . Simple Groups Additional Examples 4 / 26 References
Definitions/Theorems Sylow Theorems Lagrange’s Theorem : Let G be a finite group and let Andrew Clarey H � G . Then | H | | | G | , as | G | = | H | [ G : H ] where [ G : H ] Definitions/ Theorems is the number of Left/Right cosets. Groups, Subgroups Lagrange’s, Let H � G . Then the number of Left/Right Cosets of H Normality Class Equation, in G is [ G : H ], called the index . Cauchy’s Theorem First Sylow Let H � G . Then we say H is a normal subgroup if Theorem ∀ h ∈ H , g ∈ G , ghg − 1 ∈ H . We write H � G . Theorem Examples Proof Theorem : Let H � G . Then the following are equivalent: Additional Proofs Second Sylow Theorem i) H � G ii) gHg − 1 = H , ∀ g ∈ G Third Sylow Theorem iii) gH = Hg , ∀ g ∈ G Results Cyclic subgroups Simple Groups Additional Examples 5 / 26 References
Definitions/Theorems Sylow If H is the only subgroup in G of order | H | then H � G . Theorems Andrew Clarey If H � G then G / H is a group called the quotient group whose elements are of the form gH , ∀ g ∈ G , and whose Definitions/ Theorems operation is ∗ such that aH ∗ bH = ( a ∗ b ) H . Groups, Subgroups Lagrange’s, Normality If G , H are groups, then we can define a function φ : Class Equation, Cauchy’s Theorem G → H as a homomorphism if φ ( g 1 g 2 ) = φ ( g 1 ) φ ( g 2 ). First Sylow Theorem Define a surjection φ from G → G / H where g → gH . Theorem Examples The kernel of φ is given by Ker ( φ ) = { g ∈ G | φ ( g ) = e H } , Proof where e H is the identity in H and it is a normal subgroup. Additional Proofs Second Sylow The Normalizer of H � G is the subset Theorem N ( H ) = { g ∈ G | gHg − 1 = H } . Third Sylow Theorem The Center of a group G is the set of elements Results Cyclic subgroups Z ( G ) = { a ∈ G | ag = ga , ∀ g ∈ G } . Simple Groups Additional Examples 6 / 26 References
Definitions/Theorems Sylow Theorems Andrew Clarey The Centralizer of a g ∈ G is the set of elements Definitions/ Z ( g ) = { a ∈ G | ag = ga } Theorems Groups, Subgroups Theorem : The Class Equation of a group G states: Lagrange’s, Normality | G | = | Z ( G ) | + [ G : Z ( g 1 )] + · · · + [ G : Z ( g k )], Class Equation, Cauchy’s Theorem g 1 , . . . , g k / ∈ Z ( G ), where each g i is a representative of a First Sylow Theorem conjugacy class which contains at least 2 elements. Theorem Examples Cauchy’s Theorem : Let G be an abelian group, and let p Proof Additional Proofs be a prime such that p | | G | . Then G contains an element Second Sylow of order p . That is, ∃ x ∈ G so that p is the lowest Theorem non-zero number such that x p = e . Third Sylow Theorem Results Cyclic subgroups Simple Groups Additional Examples 7 / 26 References
First Sylow Theorem Sylow Theorems Andrew Clarey A subgroup of a group G is called a p -Sylow subgroup if Definitions/ its order is p n , p a prime and n ∈ Z + , such that p n | | G | Theorems and p n +1 ∤ | G | . Groups, Subgroups Lagrange’s, Normality Class Equation, First Sylow Theorem: Let G be a finite group, p a Cauchy’s Theorem prime, k ∈ Z + . First Sylow Theorem i) If p k | | G | , then G has a subgroup of order p k . In Theorem Examples particular, G has a p -Sylow subgroup. Proof Additional Proofs ii) Let H be any p -Sylow subgroup of G . If K � G , | K | = p k , Second Sylow then for some g ∈ G we have K ⊆ gHg − 1 . In particular, K Theorem is contained in some p -Sylow subgroup of G . Third Sylow Theorem Results Cyclic subgroups Simple Groups Additional Examples 8 / 26 References
Examples Sylow Theorems Andrew Clarey Say | G | = 2 2 · 3 4 · 5 2 · 7 2 . Then we know there will be at least Definitions/ Theorems Groups, Subgroups one of each: Lagrange’s, Normality 2-Sylow subgroup of order 4, Class Equation, Cauchy’s Theorem 3-Sylow subgroup of order 81, First Sylow 5-Sylow subgroup of order , Theorem Theorem 7-Sylow subgroup of order . Examples Proof We also know there will be subgroups of order 2, 3, 9, 27, 5, Additional Proofs and 7. Second Sylow Theorem Third Sylow Theorem Results Cyclic subgroups Simple Groups Additional Examples 9 / 26 References
Examples Sylow Theorems Let G = A 4 , a group of order 12 = 2 2 · 3 Andrew Clarey Definitions/ Theorems A 4 = { e , (1 , 2)(3 , 4) , (1 , 3)(2 , 4) , (1 , 4)(2 , 3) , (1 , 2 , 3) , (1 , 2 , 4) , Groups, Subgroups (1 , 3 , 4) , (1 , 3 , 2) , (1 , 4 , 3) , (1 , 4 , 2) , (2 , 3 , 4) , (2 , 4 , 3) } Lagrange’s, Normality Class Equation, Cauchy’s Theorem So, a 2-Sylow subgroup of G would be a subgroup of order 4, First Sylow Theorem an example is: Theorem Examples Proof Additional Proofs H = { e , (1 , 2)(3 , 4) , (1 , 3)(2 , 4) , (1 , 4)(2 , 3) } Second Sylow Theorem In fact this is the only one and therefore is normal, and all Third Sylow Theorem subgroups of order 2 and 4 are contained within it. Results Cyclic subgroups Simple Groups Additional Examples 10 / 26 References
Examples Sylow Theorems Say G = SL 2 ( Z 3 ). 1 Then | G | = 24 = 2 3 · 3 and − 1 ≡ 2mod3. Andrew Clarey The only 2-Sylow subgroup is: Definitions/ Theorems � � � � � � � � Groups, Subgroups 1 0 0 − 1 1 1 − 1 1 Lagrange’s, , , , , Normality − 1 0 1 1 0 1 1 1 Class Equation, Cauchy’s Theorem � � � � � � � � − 1 0 0 1 − 1 − 1 1 − 1 First Sylow , , , Theorem − 1 − 1 − 1 − 1 − 1 0 0 1 Theorem Examples Proof Additional Proofs and there are 4 3-Sylow subgroups: Second Sylow Theorem � � � � � � � � � � � � � � � � 1 1 1 0 0 1 0 − 1 Third Sylow , , , Theorem − 1 − 1 − 1 0 1 1 1 1 Results Cyclic subgroups Simple Groups Additional Examples 11 / 26 References
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