How large can a finite group of matrices be? Blundon Lecture UNB Fredericton 10/13/2007

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2 How large can a finite group of matrices be? Blundon Lecture UNB Fredericton 10/13/2007 Martin Lorenz Temple University, Philadelphia

3 Overview Groups... and some of their uses Martin Lorenz How large can a finite group of matrices be? slide 2

4 Overview Groups... and some of their uses The size of finite matrix groups Martin Lorenz How large can a finite group of matrices be? slide 2

5 Overview Groups... and some of their uses The size of finite matrix groups The Minkowski sequence: two mysteries Martin Lorenz How large can a finite group of matrices be? slide 2

6 Reference R. M. Guralnick & L.: Orders of finite groups of matrices, Contemp. Math. 420, (2006), arxiv:math.gr/ Martin Lorenz How large can a finite group of matrices be? slide 3

7 Reference R. M. Guralnick & L.: Orders of finite groups of matrices, Contemp. Math. 420, (2006), arxiv:math.gr/ pdf file of this talk on my web page Martin Lorenz How large can a finite group of matrices be? slide 3

8 Part I: Groups

9 The definition of a group... here it is (from Bourbaki): Martin Lorenz How large can a finite group of matrices be? slide 5

10 The definition of a group... here it is (from Bourbaki): Groups have been around long before they were defined in the above terms... Martin Lorenz How large can a finite group of matrices be? slide 5

Origins and some uses of groups Geometry: symmetry groups Given X R n (e.g., a Platonic solid in R 3 ), any distance preserving transformation of R n that maps X to itself is called a symmetry of X.

11 Origins and some uses of groups Geometry: symmetry groups Given X R n (e.g., a Platonic solid in R 3 ), any distance preserving transformation of R n that maps X to itself is called a symmetry of X. (from MacTutor History of Mathematics archive) Martin Lorenz How large can a finite group of matrices be? slide 6

12 Origins and some uses of groups Example: the dodecahedron symmetry group A 5 It has 60 symmetries which form the so-called alternating group of degree 5. Martin Lorenz How large can a finite group of matrices be? slide 6

13 Origins and some uses of groups Physics: gauge groups Theories in physics are often described by Lagrangians which are invariant under certain symmetry transformation groups, called gauge groups. These are infinite groups such as the orthogonal group O n (R) = {A M n (R) Ax = x x R n } = {A M n (R) A A tr = 1 n n } n n-matrices transpose matrix Martin Lorenz How large can a finite group of matrices be? slide 6

14 Origins and some uses of groups Algebra: Galois groups Recall: the quadratic polynomial equation ax 2 + bx + c = 0 has solutions x 1,2 = b ± b 2 4ac 2a Martin Lorenz How large can a finite group of matrices be? slide 6

15 Origins and some uses of groups Algebra: Galois groups Similar formulas (more complicated and requiring higher roots) are known for cubic and quartic polynomials. Tartaglia, Ferrari, Cardano; 16 th century Martin Lorenz How large can a finite group of matrices be? slide 6

16 Origins and some uses of groups Algebra: Galois groups Similar formulas (more complicated and requiring higher roots) are known for cubic and quartic polynomials. Tartaglia, Ferrari, Cardano; 16 th century For degree 5, however, this is no longer possible! Niels Henrik Abel, 1824 Martin Lorenz How large can a finite group of matrices be? slide 6

17 Origins and some uses of groups Algebra: Galois groups The ultimate reason for the non-solvability of the quintic equation by radicals is group theoretical: A 5 is non-abelian simple Martin Lorenz How large can a finite group of matrices be? slide 6

18 Origins and some uses of groups FINAL WORD: associated to a polynomial eq n of any degree, there is a finite group, the Galois group of the eq n. The eq n is solvable by radicals Evariste Galois its Galois group has no non-abelian simple pieces Martin Lorenz How large can a finite group of matrices be? slide 6

19 The Enormous Theorem Finite simple groups are the elementary particles of finite group theory. Martin Lorenz How large can a finite group of matrices be? slide 7

20 The Enormous Theorem Finite simple groups are the elementary particles of finite group theory. They are now all known; the Enormous Theorem (about 1983) gives a complete classification: There are several series of finite simple groups (e.g. A 5, A 6, A 7,... ) and 26 isolated ones, known as the sporadic groups. Martin Lorenz How large can a finite group of matrices be? slide 7

21 Size of the Classification Project The original proof of the Enormous Theorem was spread over a vast number journal articles and various unpublished manuscripts, some of which are incomplete. Martin Lorenz How large can a finite group of matrices be? slide 8

22 Size of the Classification Project The original proof of the Enormous Theorem was spread over a vast number journal articles and various unpublished manuscripts, some of which are incomplete. Some skepticism on the proof remains; it is currently being completely reworked. The 2 nd -generation proof will run to approximately 5,000 pages when finished. Martin Lorenz How large can a finite group of matrices be? slide 8

23 Size of the Classification Project The original proof of the Enormous Theorem was spread over a vast number journal articles and various unpublished manuscripts, some of which are incomplete. Some skepticism on the proof remains; it is currently being completely reworked. The 2 nd -generation proof will run to approximately 5,000 pages when finished. The largest sporadic group is known as the Monster; it has size Martin Lorenz How large can a finite group of matrices be? slide 8

24 Part II: Matrix Groups

25 Finite groups of matrices We will be looking at matrices in GL n (C) = {all invertible n n-matrices over C} Martin Lorenz How large can a finite group of matrices be? slide 10

26 Finite groups of matrices We will be looking at matrices in GL n (C) = {all invertible n n-matrices over C} Def n : A finite subgroup of GL n (C) is a finite (non-empty) collection of matrices G GL n (C) satisfying A,B G A B G The number of matrices in G is called the order of G. Martin Lorenz How large can a finite group of matrices be? slide 10

27 Finite groups of matrices Example: symmetries of the square (n = 2) matrices: 90 rotation: reflection across y-axis: ( ) ( ) Martin Lorenz How large can a finite group of matrices be? slide 10

28 Finite groups of matrices Example: symmetries of the square (n = 2) matrices: 90 rotation: reflection across y-axis: ( ) ( ) The group G = { all symmetries of the square } GL 2 (C) has order G = 4 2 = 2 n n!. Martin Lorenz How large can a finite group of matrices be? slide 10

29 Finite groups of matrices Basic Q n : Given n, what are the possible orders of finite subgroups of GL n (C)? Martin Lorenz How large can a finite group of matrices be? slide 10

30 Finite groups of matrices Basic Q n : Given n, what are the possible orders of finite subgroups of GL n (C)? Answer: They can be anything you want! Indeed, pick s and form the scalar matrix A = e 2πi/s... e 2πi/s n n Then G = {1 n n,a,a 2,...A s 1 } is a subgroup of GL n (C) of order s. Martin Lorenz How large can a finite group of matrices be? slide 10

31 Finite groups of matrices Modified Q n : What about GL n (Q) instead? Martin Lorenz How large can a finite group of matrices be? slide 10

32 Minkowski s Theorem geometry of numbers relativity: space-time quadratic forms Hermann Minkowski Martin Lorenz How large can a finite group of matrices be? slide 11

33 Minkowski s Theorem Minkowski (1887): The least common multiple of the orders of all finite subgroups of GL n (Q) is given by M(n) = l prime l n l 1 + n l(l 1) + n l (l 1) Here is the greatest integer ( floor ) function Martin Lorenz How large can a finite group of matrices be? slide 11

34 Minkowski s Theorem Minkowski (1887): The least common multiple of the orders of all finite subgroups of GL n (Q) is given by M(n) = l prime l n l 1 + n l(l 1) + n l (l 1) The l-factor for l > n + 1 equals 1; so the product is finite. E.g., M(1) = 2 1 = 2 M(2) = = 24 M(3) = = 48 M(4) = = 5760 Martin Lorenz How large can a finite group of matrices be? slide 11

35 Minkowski s Theorem Minkowski (1887): The least common multiple of the orders of all finite subgroups of GL n (Q) is given by M(n) = l prime l n l 1 + n l(l 1) + n l (l 1) Recursion: M(2n + 1) = 2M(2n) M(2n) = 2 M(2n 1) l prime, l 1 2n ln l Martin Lorenz How large can a finite group of matrices be? slide 11

36 Minkowski s Theorem Minkowski (1887): The least common multiple of the orders of all finite subgroups of GL n (Q) is given by M(n) = l prime l n l 1 + n l(l 1) + n l (l 1) Recursion: M(2n + 1) = 2M(2n) M(2n) = 2 M(2n 1) l prime, l 1 2n ln l }{{} = denominator of B 2n n Bernoulli numbers: x e x 1 = n 0 B nx n n! Martin Lorenz How large can a finite group of matrices be? slide 11

37 Minkowski s Theorem Minkowski (1887): The least common multiple of the orders of all finite subgroups of GL n (Q) is given by M(n) = l prime l n l 1 + n l(l 1) + n l (l 1) Using the identity (m!) l = l m l + m l , one can rewrite the l-factor: l-part M(n) l = l n l 1 ( n l 1! ) l Martin Lorenz How large can a finite group of matrices be? slide 11

38 Minkowski s Theorem Minkowski (1887): The least common multiple of the orders of all finite subgroups of GL n (Q) is given by M(n) = l prime l n l 1 + n l(l 1) + n l (l 1) More on the Minkowski numbers M(n) later... Martin Lorenz How large can a finite group of matrices be? slide 11

39 Constructing large matrix groups Example: GL n (Q) contains the subgroup of monomial matrices : exactly one entry ±1 in each row and column, all other entries are 0 O n (Z) = ±1... S n ±1 Martin Lorenz How large can a finite group of matrices be? slide 12

40 Constructing large matrix groups Example: GL n (Q) contains the subgroup of monomial matrices : exactly one entry ±1 in each row and column, all other entries are 0 O n (Z) = ±1... S n ±1 This group has order O n (Z) = 2 n n! O n (Z) 2 = M(n) 2 (n = 2: symmetries of the square) Martin Lorenz How large can a finite group of matrices be? slide 12

41 Constructing large matrix groups Example: GL n (Q) contains the subgroup of monomial matrices : exactly one entry ±1 in each row and column, all other entries are 0 O n (Z) = ±1... S n ±1 This group has order O n (Z) = 2 n n! O n (Z) 2 = M(n) 2 (n = 2: symmetries of the square) Similarly: for each prime l, there is a subgroup G GL n (Q) such that G l = M(n) l. Martin Lorenz How large can a finite group of matrices be? slide 12

42 Constructing large matrix groups Hence, the main content of Minkowski s Theorem is: The order of any finite subgroup of GL n (Q) divides M(n) Martin Lorenz How large can a finite group of matrices be? slide 12

43 Constructing large matrix groups Hence, the main content of Minkowski s Theorem is: The order of any finite subgroup of GL n (Q) divides M(n) How does one go about proving this? Martin Lorenz How large can a finite group of matrices be? slide 12

44 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Martin Lorenz How large can a finite group of matrices be? slide 13

45 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Step 1 (group theory & linear algebra): After a base change, G GL n (Z) (!) Martin Lorenz How large can a finite group of matrices be? slide 13

46 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Step 1 (group theory & linear algebra): After a base change, G GL n (Z) (!) So we can reduce entries modulo another prime p : GL n (Z) GL n (Z/pZ) A = (a ij ) A = (a ij mod p) Martin Lorenz How large can a finite group of matrices be? slide 13

47 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Step 1 (group theory & linear algebra): After a base change, G GL n (Z) (!) So we can reduce entries modulo another prime p : GL n (Z) GL n (Z/pZ) A = (a ij ) A = (a ij mod p) As long as p 2, this map is 1-to-1 on G (!) Hence G GL n (Z/pZ). Martin Lorenz How large can a finite group of matrices be? slide 13

48 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Step 1 (group theory & linear algebra): After a base change, G GL n (Z) (!) So we can reduce entries modulo another prime p : GL n (Z) GL n (Z/pZ) A = (a ij ) A = (a ij mod p) As long as p 2, this map is 1-to-1 on G (!) Hence G GL n (Z/pZ). In fact, by Lagrange s Theorem (!) G divides GL n (Z/pZ) Martin Lorenz How large can a finite group of matrices be? slide 13

49 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Counting bases of n-space over Z/pZ one obtains GL n (Z/pZ) = (p n 1)(p n p)(p n p 2 )...(p n p n 1 ) n = p n(n 1)/2 (p i 1) i=1 Martin Lorenz How large can a finite group of matrices be? slide 13

50 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Counting bases of n-space over Z/pZ one obtains GL n (Z/pZ) = (p n 1)(p n p)(p n p 2 )...(p n p n 1 ) n = p n(n 1)/2 (p i 1) i=1 To summarize: if p 2 then the order G divides this number; so p 2,p l G l n i 1 (pi 1) l Martin Lorenz How large can a finite group of matrices be? slide 13

51 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Step 2 (number theory): l 2 n i 1 (pi 1) l = M(n) l for infinitely many primes p Martin Lorenz How large can a finite group of matrices be? slide 13

52 Idea of proof Given: G a finite subgp. of GL n (Q) l a prime Want: G l divides M(n) l Step 2 (number theory): l 2 n i 1 (pi 1) l = M(n) l for infinitely many primes p G l M(n) l at least for l 2 The prime l = 2 requires more work, as usual... Martin Lorenz How large can a finite group of matrices be? slide 13

53 Part III: Two Mysteries

54 Minkowski s sequence M(n) Entering the first six terms of M(n), 2, 24, 48, 5760, 11520, into the On-Line Encyclopedia of Integer Sequences by Neil Sloane (AT&T) brings up sequence A Martin Lorenz How large can a finite group of matrices be? slide 15

55 Minkowski s sequence M(n) A has two alternative descriptions... Martin Lorenz How large can a finite group of matrices be? slide 15

56 Sequence M(n) First Alternative Integer-valued polynomials on primes: Consider all polynomials f(x) Q[x] of degree n such that f(p) Z for all primes p. Martin Lorenz How large can a finite group of matrices be? slide 16

57 Sequence M(n) First Alternative Integer-valued polynomials on primes: Consider all polynomials f(x) Q[x] of degree n such that f(p) Z for all primes p. The smallest positive leading coefficient of any such polynomial is a fraction of the form 1 for some positive integer a(n). a(n) Martin Lorenz How large can a finite group of matrices be? slide 16

58 Sequence M(n) First Alternative Integer-valued polynomials on primes: Consider all polynomials f(x) Q[x] of degree n such that f(p) Z for all primes p. The smallest positive leading coefficient of any such polynomial is a fraction of the form 1 for some positive integer a(n). a(n) Minkowski s formula is identical with the one proved independently for a(n + 1) by Chabert et. al.; so a(n + 1) = M(n) Reference: Jean-Luc Chabert, Scott T. Chapman, and William W. Smith, A basis for the ring of polynomials integer-valued on prime numbers, Factorization in integral domains (Iowa City, IA, 1996), Lecture Notes in Pure and Appl. Math., vol. 189, Dekker, New York, 1997, pp Martin Lorenz How large can a finite group of matrices be? slide 16

59 Sequence M(n) Second Alternative Logarithm power expansion: Let P(n,z) Q[z] be the coefficient of x n in the Taylor series for ( ln(1 x) x ) z. Paul Hanna: OEIS Martin Lorenz How large can a finite group of matrices be? slide 17

60 Sequence M(n) Second Alternative Logarithm power expansion: Let P(n,z) Q[z] be the coefficient of x n in the Taylor series for ( ln(1 x) x ) z. Paul Hanna: OEIS In detail: put ξ = ln(1 x) x ( ln(1 x) x ) z = (1 + ξ) z = 1 = k=1 ( ) z m m 0 x k k+1 to get ξ m = 1 + n 1 P(n,z)x n Martin Lorenz How large can a finite group of matrices be? slide 17

61 Sequence M(n) Second Alternative Logarithm power expansion: Let P(n,z) Q[z] be the coefficient of x n in the Taylor series for ( ln(1 x) x ) z. Paul Hanna: OEIS In detail: put ξ = ln(1 x) x ( ln(1 x) Examples: x ) z = (1 + ξ) z = 1 = k=1 ( ) z m m 0 x k k+1 to get ξ m = 1 + n 1 P(n,z)x n P(1,z) = z P(2,z) = 5z+3z P(3,z) = 6z+5z2 +z 3 P(4,z) = 502z+485z2 +150z 3 +15z Martin Lorenz How large can a finite group of matrices be? slide 17

62 Sequence M(n) Second Alternative Logarithm power expansion: Let P(n,z) Q[z] be the coefficient of x n in the Taylor series for ( ln(1 x) x ) z. Paul Hanna: OEIS Putting b(n) = denominator of P(n,z), it has recently been proved that b(n) = a(n + 1) So b(n) = M(n). Reference: Jean-Luc Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European Journal of Combinatorics 28 (2007), Martin Lorenz How large can a finite group of matrices be? slide 17

63 Coincidence? Martin Lorenz How large can a finite group of matrices be? slide 18

64 ... if time... if time

65 Large matrix groups (again)... if time Recall: GL n (Q) contains the group of monomial matrices O n (Z) of order O n (Z) = 2 n n! Martin Lorenz How large can a finite group of matrices be? slide 20

66 Large matrix groups (again)... if time Recall: GL n (Q) contains the group of monomial matrices O n (Z) of order O n (Z) = 2 n n! Feit (unpubl., 1998): O n (Z) = 2 n n! is the largest order of any finite subgroup of GL n (Q) for n > 10 and n = 1, 3, 5. Moreover, O n (Z) is the unique subgroup of that order, up to conjugacy. Walter Feit Martin Lorenz How large can a finite group of matrices be? slide 20

67 Large matrix groups (again)... if time Feit relies on an unfinished manuscript of Weisfeiler on the Jordan bound (based on the Enormous Theorem). Boris Weisfeiler has been missing in Chile since January 4, 1985 ( Martin Lorenz How large can a finite group of matrices be? slide 20

68 Large matrix groups (again)... if time Feit relies on an unfinished manuscript of Weisfeiler on the Jordan bound (based on the Enormous Theorem). Weisfeiler s work has now been completed (and improved) by Michael Collins (preprints, Oxford University, 2005) Martin Lorenz How large can a finite group of matrices be? slide 20

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