On the classication of algebras

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Technische Universität Carolo-Wilhelmina Braunschweig Institut Computational Mathematics On the classication of algebras Morten Wesche September 19, 2016

Introduction Higman (1950) published the papers Enumerating p-groups I and II and introduced the so-called PORC-functions. (PORC stands for Polynomial On Residue Classes.) A function f is called PORC if there exists a natural number N and polynomials f 0,..., f N 1 such that f(x) f a (x) for all x N Z + a. So, f behaves like a polynomial for all x in the same residue class modulo N. Higman (1950) proofed that the number f d (q) of isomorphism classes of algebras of xed dimension d over an arbitrary eld with q elements can be described by a PORC-function in q. He conjectured that the number g n (p) of isomorphism types of p-groups of order p n is given by a PORC-function in p. It is a still open problem in group theory. Vaughan-Lee (2013) sketched a method to compute the PORC-functions for f d (q) and he determined them for d 4. This paper will give an introduction to Higman's PORC theory and the basic ideas of the classication of nite dimensional algebras over nite elds. Contents 1 Basic denitions and examples 2 2 Enumerating algebras 4 3 Improvement of the lemma of Burnside, Frobenius, and Cauchy 5 4 The PORC-conjecture 6

1 Basic denitions and examples Denition 1.1: Algebra An algebra A of dimension d over a eld K is a d-dimensional vector space over K equipped with a bilinear mapping K d K d K d called multiplication. If the multiplication is associative 1, then A is called associative. If the multiplication is Jacobian 2, then A is called a Lie-algebra. Remark 1.2 It is not assumed that an algebra contains an identity element. Example 1.3 1. Let V be any nite dimensional vector space over any eld K. Equipping V with the trivial multiplication V V V, (x, y) 0, gives an algebra. It is easy to see, that it is an associative and commutative algebra. 2. Let V M n (K) be the vector space of all n n matrices with entries in the eld K. Equipping V with the standard multiplication of matrices, one gets an associative algebra. It is not commutative. 3. Let V R 3 equipped with the cross product: : R 3 R 3 R 3, (x, y) x y. As this multiplication is Jacobian, (R 3, ) is a Lie-algebra. There are a lot of applications for this algebra: In maths it can be used for the calculation of the distance of skew lines or the volume of polyhedra. In physics it is used for computing an angular momentum or the Lorentz force. Denition 1.4: f d (K) Let f d (K) be the number of isomorphism types of algebras of dimension d over the eld K. Remark 1.5 Two algebras are isomorphic if there exists a bijective base transformation preserving their structure. Central Question: What can be said about f d (K) depending on d and K? Example 1.6 Independently from the chosen eld, there are two isomorphism types of one dimensional algebras: The algebra A 1 with zero multiplication : A 1 A 1 A 1, (x, y) 0, and the algebra A 2 with non-zero multiplication. When increasing the dimension to two, there are already innitely many algebras 3 if the eld is not nite (see [Goze and Remm, 2011]). However, if the eld is nite with K p e q, then the number of isomorphism types can be given by polynomials (see [Vaughan-Lee, 2013]). d 1 d 2 K 2 q 4 + q 3 + 4q 2 + 3q + 6, p 2 K p e q 2 q 4 + q 3 + 4q 2 + 4q + 6, p 3 q 4 + q 3 + 4q 2 + 4q + 7, p 5 Table 1: Number of isomorphism types of algebras for small dimension d 1 Associative: for all a, b, c A holds (ab)c a(bc). 2 Jacobian: for all a, b, c A holds (ab)c + (bc)a + (ca)b 0. 3 For instances, take the algebras with basis A k a, b a 2 0, ab 0, ba b, b 2 ka for an element k K \ {0}. These algebras are pairwise non-isomorphic. 2

Remark 1.7 For a nite eld K with K p e q write f d (K) f d (q). Theorem 1.8 (Higman) For a xed d N the function f d (q) is PORC. Denition 1.9: PORC A function f is called PORC (Polynomial On Residue Classes), if there exists a natural number N N and a set of polynomials f 0,..., f N 1 such that f(x) f a (x) for all x N Z + a. Remark 1.10 A function, that is PORC modulo N, behaves like a polynomial for all x of the same residue class modulo N. Example 1.11 Let p 5 be a prime. How many irreducible polynomials of degree three of the form x 3 c exist in F p [x], where F p is the nite eld with p elements? In other words: How many elements c F p exist, such that the polynomial x 3 c has no roots in F p? It is obvious, that c 0 must hold. Therefore, one can assume to choose c from the unit group F p. Therefore, this group shall be looked at in more detail. First of all, x an element ϑ ϑ p which generates this unit group of F p, hence it is ϑ F p. Its order is p 1 and one can nd an integer k with 0 k < p 1 such that c ϑ k. Next, dene the subgroup H ϑ 3 F p. Depending on p, H can or cannot be a trivial subgroup. Additionally, as x 0 cannot be a root of x 3 c, one can write x ϑ l with 0 l < p 1. So, ndig the roots of x 3 c 0 is equivalent to nding all solutions of ϑ 3l ϑ k. 1. Let p 1 mod 3. Therefore, the subgroup H is a proper subgroup of index 3 in F p. With x 3 ϑ 3l H the equation 3l k can just be solved with k 0 mod 3. There are (p 1)/3 such elements ϑ k. As there is a total of p 1 polynomials of the form x 3 c with c 0 and as there are (p 1)/3 reducible polynomials, there remain 2(p 1)/3 irreducible polynomials of the desired form. 2. Let p 2 mod 3. Then it is H F p So, for any l, there is a k such that the equation ϑ 3l ϑ k holds. So the polynomial x 3 c F p [x] is always reducible and there are no irreducible polynomials of the desired form. Depending on the residue class of p modulo 3 one has two dierent polynomials. PORC. Putting all this together, one has Therefore, the number is {f(x) x 3 c F p [x] f(x) is irreducible} { 2(p 1)/3, p 1 mod 3 0, p 2 mod 3 gcd(p 1, 3) 1 (p 1). 3 3

2 Enumerating algebras Remark 2.1 For the rest of this paper let K always be a nite eld with K p e q. Proposition and Denition 2.2: Structure constants The multiplication of a d-dimensional algebra with basis B {b 1,..., b d } is completely determined by a set of structure constants λ ijk K, 1 i, j, k d, via b i b j d λ ijk b k. k1 Example 2.3 1. In case of an algebra with zero multiplication, it is λ ijk 0 for all i, j, k. 2. In the case of the matrix algebra M n (K) rst x a basis: Let this be B {e ij 1 i, j n}, where e ij denotes that n n matrix, which has a one in row i at column j and whose other entries are zero. It is known that e ij e kl δ jk e il holds. Here, δ jk denotes the Kronecker delta, which evaluates to one, if both indices are equal, and which evaluates to zero otherwise. Then, using the double indices, the structure constants are { 1, if i 2 j 1, i 1 k 1, j 2 k 2, λ i1i 2, j 1j 2, k 1k 2 0, otherwise. 3. The structure constants of the algebra (R 3, ) are given as follows (when using the canonical vector space base {e 1, e 2, e 3 }: λ 123 λ 312 λ 231 1, λ 132 λ 213 λ 321 1, λ ijk 0 otherwise. Writing a set of structure constants as a vector v K d3 (the elements λ ijk are arranged lexicographically), then the action of a basis transformation G GL(d, K) is given by v G G G 1, where stands for the Kronecker product. The number of isomorphism types is therefore equal to the number of orbits of elements of K d3 under the action of GL(d, K). Using the lemma of Burnside, Cauchy, and Frobenius, one gets f d (q) 1 GL(d, F q ) 1 GL(d, F q ) conjugacy classes cl GL(d,F q) g GL(d,F q) Fix g conjugacy classes cl GL(d,F q) cl Fix g(cl) C GL(d,K) (g(cl)) Fix g(cl). It is Fix g {v K d3 v G G G 1 v}, the length of a class cl is denoted by cl, a representative of cl is given by g(cl), and C GL(d,K) (g(cl)) {x G gx xg} is the centraliser of the element g g(cl) in GL(d, K). Remark 2.4 Fix g is the eigenspace of G G G 1 belonging to the eigenvalue 1. Problem: The conjugacy classes still depend on q. While increasing q, the number of conjugacy classes of GL(d, K) increases, too. 4

3 Improvement of the lemma of Burnside, Frobenius, and Cauchy The main work of improving the application of the lemma is done by Charles Green, Graham Higman, and Michael Vaughan-Lee. Green published the paper The characters of the nite general linear groups (see [Green, 1955]) and there he dened the so called type of a matrix. This classication is based on the rational canonical form of a matrix, but is far coarser than the classication into conjugacy classes. The most important result is that the types do not depend on the eld. Graham Higman published his very well known papers Enumerating p-groups I and II (see [Higman, 1960a], [Higman, 1960b]). Although working with p-groups, he gives some theorems for algebras too. For instance, he proofed that the number of isomorphism types of algebras of xed dimension over a nite eld is PORC. Michael Vaughan-Lee then developed algorithms to really determine the PORC functions. In his paper Enumerating algebras over a nite eld (see [Vaughan-Lee, 2013]) he publishes the PORC functions for algebras of dimension d 4. (In the case d 2 the polynomials are of degree 4, for d 3 they are of degree 18, and for d 4 they are of degree 48.) The used algorithms are described in his paper Choosing elements from nite elds (see [Vaughan-Lee, 2012a]). Charles Alexander Green 4 (26 February 1926 7 April 2014) Graham Higman 5 (19 January 1917 8 April 2008) Michael Vaughan-Lee 6 Dates Work by Green: Fix g(cl) does not depend on g(cl), but on the type of the class only. The number of dierent types within GL(d, K) depends on d only, not on q. Work by Higman: Fix g(cl) is a polynomial in q. C GL(d,K) (g(cl)) is a polynomial in q. The number of elements of a xed type in GL(d, K) is PORC. Work by Vaughan-Lee: Development of algorithms to determine expicitly the PORC functions in small cases of d. 4 Picture taken from https://static.independent.co.uk/s3fs-public/styles/story_large/public/thumbnails/image/ 2014/07/20/19/sandy-green.jpg 5 Picture taken from https://upload.wikimedia.org/wikipedia/commons/2/2d/graham_higman.jpg 6 Picture taken from https://www.lincoln.ac.uk/home/media/universityoflincoln/schoolofmathematicsandphysics/ Professor-Michael-Vaughan-Lee.jpg 5

4 The PORC-conjecture When looking at Higman's papers Enumerating p-groups I and II one gets directly another possible question: How many isomorphism types of groups of order p n exist? Let the number of those isomorphism classes be g n (p). Higman formulated his famous PORC conjecture dealing with this number. Conjecture 4.1 (Higman's PORC conjecture) For xed n N, the function g n (p) giving the number of isomorphism types of groups of order p n is PORC. The problem is solved for n 7. For instance, those polynomials can be found at http://groupprops.subwiki. org/wiki/higman's_porc_conjecture or in [Vaughan-Lee, 2012b] (for n 5). However, it is an still open problem for n 8. References [Goze and Remm, 2011] Goze, M. and Remm, E. (2011). 2-dimensional algebras. Afr. J. Math. Phys., 10(1):81 91. [Green, 1955] Green, J. A. (1955). The characters of the nite general linear groups. Trans. Amer. Math. Soc., 80:402447. [Higman, 1960a] Higman, G. (1960a). Enumerating p-groups. I. Inequalities. Proc. London Math. Soc. (3), 10:2430. [Higman, 1960b] Higman, G. (1960b). Enumerating p-groups. II. Problems whose solution is PORC. London Math. Soc. (3), 10:566582. Proc. [Vaughan-Lee, 2012a] Vaughan-Lee, M. (2012a). Choosing elements from nite leds. [Vaughan-Lee, 2012b] Vaughan-Lee, M. (2012b). Graham Higman's PORC conjecture. Jahresber. Dtsch. Math.- Ver., 114(2):89106. [Vaughan-Lee, 2013] Vaughan-Lee, M. (2013). Enumerating algebras over a nite eld. Int. J. Group Theory, 2(3):4961. 6

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