The Monster Group

Historybull Discovery of the Monster came out of the

search for new finite simple groups during the 1960s and 70s

Steps to discovering the Monster1 Eacutemile Mathieursquos discovery of the group of

permutations M242 discovery of the Leech Lattice in 24

dimensions3 JH Conwayrsquos discovery of Co1 (Conwayrsquos

largest simple group)4 Bernd Fischerrsquos discovery of the Monster

Overview of the Monster

bull Largest sporadic group of the finite simple groups

bull Order = 246 320 59 76 112 133 17 19 23

29 31 41 47 59 71= 808017424794512875886459904961710757005754368000000000

bull Smallest of dimensions which the Monster can act nontrivially = 196883

Overview continued

bull Characteristic table is 194 x 194

bull Contains at least 43 conjugacy classes of maximal subgroups

bull 19 of the other 26 sporadic groups are subgroups of the Monster

Griessrsquos Construction of the Monster

bull 2 cross sections of the monster Conwayrsquos largest simple group (requiring 96308 dimensions) and the Baby Monster

Simple group acting on the Monster splits the space into three subspaces of the following dimensions

98304 + 300 + 98280 = 196884

Construction (contd)

bull 98304 = 212 24 = space needed for the cross-section

bull 300 = 24 + 23 + 22 + hellip+ 2 + 1 = triangular arrangement of numbers with 24 in the first row 23 in the second row etc

bull 98280 = 1965602 which comes from the Leech Lattice where there are 196560 points closest to a given point and they come in 98280 pairs

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Historybull Discovery of the Monster came out of the

search for new finite simple groups during the 1960s and 70s

Steps to discovering the Monster1 Eacutemile Mathieursquos discovery of the group of

permutations M242 discovery of the Leech Lattice in 24

dimensions3 JH Conwayrsquos discovery of Co1 (Conwayrsquos

largest simple group)4 Bernd Fischerrsquos discovery of the Monster

Overview of the Monster

bull Largest sporadic group of the finite simple groups

bull Order = 246 320 59 76 112 133 17 19 23

29 31 41 47 59 71= 808017424794512875886459904961710757005754368000000000

bull Smallest of dimensions which the Monster can act nontrivially = 196883

Overview continued

bull Characteristic table is 194 x 194

bull Contains at least 43 conjugacy classes of maximal subgroups

bull 19 of the other 26 sporadic groups are subgroups of the Monster

Griessrsquos Construction of the Monster

bull 2 cross sections of the monster Conwayrsquos largest simple group (requiring 96308 dimensions) and the Baby Monster

Simple group acting on the Monster splits the space into three subspaces of the following dimensions

98304 + 300 + 98280 = 196884

Construction (contd)

bull 98304 = 212 24 = space needed for the cross-section

bull 300 = 24 + 23 + 22 + hellip+ 2 + 1 = triangular arrangement of numbers with 24 in the first row 23 in the second row etc

bull 98280 = 1965602 which comes from the Leech Lattice where there are 196560 points closest to a given point and they come in 98280 pairs

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Overview of the Monster

bull Largest sporadic group of the finite simple groups

bull Order = 246 320 59 76 112 133 17 19 23

29 31 41 47 59 71= 808017424794512875886459904961710757005754368000000000

bull Smallest of dimensions which the Monster can act nontrivially = 196883

Overview continued

bull Characteristic table is 194 x 194

bull Contains at least 43 conjugacy classes of maximal subgroups

bull 19 of the other 26 sporadic groups are subgroups of the Monster

Griessrsquos Construction of the Monster

bull 2 cross sections of the monster Conwayrsquos largest simple group (requiring 96308 dimensions) and the Baby Monster

Simple group acting on the Monster splits the space into three subspaces of the following dimensions

98304 + 300 + 98280 = 196884

Construction (contd)

bull 98304 = 212 24 = space needed for the cross-section

bull 300 = 24 + 23 + 22 + hellip+ 2 + 1 = triangular arrangement of numbers with 24 in the first row 23 in the second row etc

bull 98280 = 1965602 which comes from the Leech Lattice where there are 196560 points closest to a given point and they come in 98280 pairs

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Overview continued

bull Characteristic table is 194 x 194

bull Contains at least 43 conjugacy classes of maximal subgroups

bull 19 of the other 26 sporadic groups are subgroups of the Monster

Griessrsquos Construction of the Monster

bull 2 cross sections of the monster Conwayrsquos largest simple group (requiring 96308 dimensions) and the Baby Monster

Simple group acting on the Monster splits the space into three subspaces of the following dimensions

98304 + 300 + 98280 = 196884

Construction (contd)

bull 98304 = 212 24 = space needed for the cross-section

bull 300 = 24 + 23 + 22 + hellip+ 2 + 1 = triangular arrangement of numbers with 24 in the first row 23 in the second row etc

bull 98280 = 1965602 which comes from the Leech Lattice where there are 196560 points closest to a given point and they come in 98280 pairs

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Griessrsquos Construction of the Monster

bull 2 cross sections of the monster Conwayrsquos largest simple group (requiring 96308 dimensions) and the Baby Monster

Simple group acting on the Monster splits the space into three subspaces of the following dimensions

98304 + 300 + 98280 = 196884

Construction (contd)

bull 98304 = 212 24 = space needed for the cross-section

bull 300 = 24 + 23 + 22 + hellip+ 2 + 1 = triangular arrangement of numbers with 24 in the first row 23 in the second row etc

bull 98280 = 1965602 which comes from the Leech Lattice where there are 196560 points closest to a given point and they come in 98280 pairs

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Construction (contd)

bull 98304 = 212 24 = space needed for the cross-section

bull 300 = 24 + 23 + 22 + hellip+ 2 + 1 = triangular arrangement of numbers with 24 in the first row 23 in the second row etc

bull 98280 = 1965602 which comes from the Leech Lattice where there are 196560 points closest to a given point and they come in 98280 pairs

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Alternate Construction

bull Let V be a vector space and dim V = 196882 over a field with 2 elements

bull Choose H subset of M st H is maximal sgH = 31+122Suz2 = one of the max subgroups of the Monster(Suz = Suzuki group)

bull Elements of monster = words in elements of H and an extra generator T

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Alternate Construction 2bull Theorem There is an algebra isomorphism between

(B ) and B This isomorphism is an isometry up to a scalar multiple and it transforms C to the group of automorphisms of B and σ to the automorphism of B

bull (B ) (x y) = πo(πox times πoy) where xy є B and πo is the orthogonal projection map of B

bull B = algebra of Griessbull F1 = ltCσgtbull C group with structure 21+24 (1)bull σ is an involutive linear automorphismNote F1 acts as an automorphism on (B ) F1 is the Monster group

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Finding the Generators of the Monster Group

Standard generators of M are a and b st a є class 2A b є class 3B o(ab) = 29

Find an element of order 34 38 50 54 62 68 94 104 or 110 This powers up to x in class 2A o(x) = 2

Find an element of order 9 18 27 36 45 or 54 This powers up to y in class 3B o(y) = 3

Find a conjugate a of x and a conjugate b of y such that ab has order 29 o(xy) = 29

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

Moonshine connectionsthe Monsterrsquos connection with number theory

1 The j-functionj(q) = q-1 + 196884q + 21493760q2 + 864299970q3 +

20245856256q4 + hellipCharacter degrees for the Monster

11968832129687684260932618538750076

Let

= m1

= m2

= m3

= m4

= m5

Let the coefficients of the j-function = j1 j2 j3 j4 j5 respectively

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

j2 = m1 + m2

j3 = m1 + m2 + m3

j4 = m1 + m1 + m2 + m2 + m3 + m4

2 J-function and Modular Theoryall prime numbers that could be used to obtain other j-functions

2 3 5 7 11 13 17 19 23 29 31 41 47 59 71= prime numbers that factor the order of the Monster

Rationalemodular group (allows one pair of integers to change into another)

operates on the hyperbolic plane surface is a sphere when the number is one of the primes above otherwise it would be a torus double torus etc

Moonshine Module infinite dimensional space having the Monster as its symmetry group which gives rise to the j-function and mini-j-functions (Hauptmoduls)

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

3 String Theorynumber of dimensions for String Theory is either 10 or 26bull a path on which time-distance is always zero in a higher

dimensional (gt 4) space-time (Lorentzian space) yields a perpendicular Euclidean space of 2 dimensions lowerex 26-dimensional Lorentzian space yields the 24-dimensional Euclidean space which contains the Leech Lattice

Leech Lattice contains a point (01234hellip232470)time distance from origin point in Lorentzian space

0 = 0sup2 + 1sup2 + 2sup2 + hellip + 23sup2 + 24sup2 - 70sup2 this point lies on a light ray through the origin

Borcherd said a string moving in space-time is only nonzero if space-time is 26-dimensional

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13

4 another connection with number theorySome special properties of the number 163

a eπradic(163) = 26253741264076874399999999999925 which is very close to a whole numberb xsup2 - x + 41 = 0 has radic(163) as one of its factors

xsup2 - x + 41 gives the prime numbers for all values of x between 1 and 40

Monster 194 columns in characteristic table which give functions163 are completely independent

ldquoUnderstanding [the Monsterrsquos] full nature is likely to shed light on the very fabric of the universerdquo Mark Ronan

• The Monster Group
• History
• Overview of the Monster
• Overview continued
• Griessrsquos Construction of the Monster
• Construction (contd)
• Alternate Construction
• Alternate Construction 2
• Finding the Generators of the Monster Group
• Moonshine connections the Monsterrsquos connection with number theory
• Slide 11
• Slide 12
• Slide 13