Frobenius groups, near-rings, and combinatorial designs (Life after Coolville) A combinatorial design is an incidence structure of points and blocks. Blocks are simply sets of points, and we say a point p is incident to a block B if p B. Design theory concentrates on the construction of designs with pre-determined properties. For example, several recent publications describe construction techniques for (100, 45, 20) symmetric designs designs with 100 points and 100 blocks, in which every block contains 45 points and every pair of blocks share 20 points. Assume that a finite group G is a semi-direct product of N by H (G = N H). We can view H as a group of automorphisms on N via h(n) = n h = hnh 1. H is fixed-point-free if h(n) = n implies n = 1 or h = 1. If the action of H is fixed-point-free, then G is called a Frobenius group. A near-ring is a set, P, with two binary operations: (P, +, ). Addition (+) defines a group on P, while multiplication ( ) defines a semigroup on P. Furthermore, addition is right-distributive over multiplication: (a + b) c = a c + b c. A near-ring is planar if two lines define a unique point; that is, if x a = x b + c has a unique solution x P. Design constructions depend on the identification of automorphism groups; thus, Frobenius groups seem to be critical for constructing large non-abelian designs. On the other hand, every Frobenius group can be used to construct a planar near-ring. I will give a very brief introduction to each topic, then indicate how the three theories might be unified a major goal of algebraic combinatorics.
(Math Sem., 4/8/04 : 1) Algebraic Combinatorics Constructive Deductive Tools: {groups, rings, modules, representations} Tools: {designs, graphs, codes} Goals: {designs, graphs, codes} Goals: {groups, rings, modules, representations} Journals: Journals: {J Combinatorial Designs, Trans. Information Theory,... } { J. Algebra, J. Combinatorial Theory,... }
(Math Sem., 4/8/04 : 2) (v, k, λ, b, r) BIBD s (balanced incomplete block designs) e x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 e 0 1 0 1 1 1 0 0 0 1 0 x 0 0 1 0 1 1 1 0 0 0 1 x 2 1 0 0 1 0 1 1 1 0 0 0 x 3 0 1 0 0 1 0 1 1 1 0 0 x 4 0 0 1 0 0 1 0 1 1 1 0 x 5 0 0 0 1 0 0 1 0 1 1 1 x 6 1 0 0 0 1 0 0 1 0 1 1 x 7 1 1 0 0 0 1 0 0 1 0 1 x 8 1 1 1 0 0 0 1 0 0 1 0 x 9 0 1 1 1 0 0 0 1 0 0 1 x 10 1 0 1 1 1 0 0 0 1 0 0 Parameters of (v, k, λ, b, r) BIBD: v = number of points (columns); k = weight of each block (row); λ = intersections between each pair of blocks; b = number of blocks(rows); r = number of blocks containing each point; Example is symmetric: b = v and r = λ. Symmetric matrix equation: DD T = (k λ)i + λj Comment: nice BIBD s can be viewed as finite geometries...
(Math Sem., 4/8/04 : 3) Automorphism groups of symmetric designs e x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 eδ 0 1 0 1 1 1 0 0 0 1 0 xδ 0 0 1 0 1 1 1 0 0 0 1 x 2 δ 1 0 0 1 0 1 1 1 0 0 0 x 3 δ 0 1 0 0 1 0 1 1 1 0 0 x 4 δ 0 0 1 0 0 1 0 1 1 1 0 x 5 δ 0 0 0 1 0 0 1 0 1 1 1 x 6 δ 1 0 0 0 1 0 0 1 0 1 1 x 7 δ 1 1 0 0 0 1 0 0 1 0 1 x 8 δ 1 1 1 0 0 0 1 0 0 1 0 x 9 δ 0 1 1 1 0 0 0 1 0 0 1 x 10 δ 1 0 1 1 1 0 0 0 1 0 0 An automorphism of a design is a permutation of the point set (column set) which preserves the block set (row set). An automorphism group, G, of a design is point-regular if, for every pair (p, q) of points, there is a unique g G which maps p to q. This is usually called a Singer group. Example: The group {f m : m = 0, 1,..., 10} = Z 11, where f m : x x 1+m, is a Singer group of this design. If D is a symmetric design admitting a Singer group, G, then D is completely determined by a polynomial δ ZG. δ is called a difference set.
(Math Sem., 4/8/04 : 4) Genuinely non-abelian symmetric designs A symmetric design, D, is genuinely non-abelian if: (1) D admits a non-abelian Singer group, and (2) no equivalent design admits an Abelian Singer group. Only known genuinely non-abelian symmetric designs are Menon-Hadamard: Examples: (100, 45, 20) [Smith, 1995]; (100, 45, 20) [Golemac and Vucicic, 2001]. (4u 2, 2u 2 + u, u 2 u) ; u = a multiple of 2, 3, or 5. Theorem [McFarland, 1989]: If p > 3 is prime, then any (4p 2, 2p 2 p, p 2 p) symmetric design with a Singer group is genuinely non-abelian. Theorem [Iiams, 1995]: Suppose G is a Singer group for a (4p 2, 2p 2 p, p 2 p) symmetric design, with p > 3 prime. G must be one of five specific groups (or six groups if p 1 mod 4).
(Math Sem., 4/8/04 : 5) Frobenius groups I A finite group, G, is a semidirect product of N by H (G = N H) if: N is a normal subgroup of G; H is a subgroup of G; G = NH = {nh : n N, h H}; N H = {e}. If G = N H, then H acts on N by conjugation: h(n) = n h = hnh 1. H is a fixed-point-free automorphism group of N if h(n) = n implies n = 1 or h = 1. Assume that G is a finite semidirect product of N by H. If the action of H on N is fixed-point-free, then G is called a Frobenius group.
(Math Sem., 4/8/04 : 6 ) Frobenius groups II Assume p is prime, q (p 1), and u has multiplicative order q mod p. F p,q is the Frobenius group: F p,q = a, b : a p = b q = 1, b 1 ab = a u. Proposition: Suppose that G is a group of order pq, where p and q are primes with p > q. Either G is Abelian, or q divides p 1 and G = F p,q. Example: F 13,4 = x, z x 13 = z 4 = 1; zxz 1 = x 5 = x x 13 = 1 z z 4 = 1 = Z13 Z 4 Suppose z m (x k ) = z m x k z m = ( x k) 5 m = x k ; then k 5 m k mod 13 and k(1 5 m ) 0 mod 13. Thus, k is a multiple of 13 or m is a multiple of 4. The action of z is fixed-point-free. Example: There are sixteen groups of order 676. Two of these are Frobenius groups: (Z 13 Z 13 ) Z 4. One is a Frobenius group: Z 169 Z 4.
Possible (4p 2, 2p 2 p, p 2 p) Singer groups (p = 13) (Math Sem., 4/8/04 : 7 ) < x, y x p = y p = 1, xy = yx > < z z 4 = 1 > Group zxz 1 zyz 1 Normal (N) Quotient (H) Comp. Quotient (G/Core(H)) G 11 x y 1 y = Z p x, z = Z 4p G 11 / x, z 2 = D 2p G 13 y 1 x N = Z p H = F p,4 G 13 /core(h) = F p,4 G 14 x y f y = Z p x, z = Z 4p G 14 / x = F p,4 G 15 x 1 y f x = Z p y, z = F p,4 G 15 / y, z 2 = D 2p G 16 x f y f x = Z p y, z = F p,4 G 16 / y = F p,4 < x x p2 = 1 > < z z 4 = 1 > Group zxz 1 Normal (N) Quotient (H) Comp. Quotient (G/Core(H)) G 4 x f x 13 = Z p F p,4 G 4 (f 2 1 mod p 2 f = 70) Proposition [Mendes, 2003]: G 11 and G 14 cannot be Singer groups when p = 13. Consequently, if G is a Menon-Hadamard Singer group for p = 13, then G = Z 13 F 13,4.
(Math Sem., 4/8/04 : 8) Near-rings I A right near-ring is a set, P, with two binary operations, (P, +, ), such that: (P, +) forms a group; (P, ) forms a semigroup ( is associative); (a + b) c = a c + b c. Note: (P, +, ) is a ring if addition is commutative and the distributive property is 2-sided. Elements a, b of a near-ring P are equivalent if pa = pb for every p P. A near-ring with at least three equivalence classes is planar if two lines define a unique point; that is, if x a = x b + c has a unique solution x P, for every inequivalent pair (a, b). Example [Fuchs, p. 648]: Let V be a normed vector space over R. Let + be vector addition. Define x y as scaling x by y: x y = y x. a b a = b. ( classes are circles about the origin.) (V, +, ) is a right planar near-ring; it is NOT a ring.
(Math Sem., 4/8/04 : 9) Example: Let D(R) be the set of differentiable functions on R. Define (f + g)(x) = f(x) + g(x). Define (f g)(x) = f(g(x)). (D(R), +) is a group; Near-rings II (D(R), ) is a semigroup, since (f g) h = f (g h); (f + g) h = f h + g h; h (f + g) h f + h g; Thus, (D(R), +, ) is a right near-ring, but not a ring. Example [Dickson, 1905]: The multiplication in a finite field can be distorted to produce a near-field a structure which obeys all the axioms for a field, except for commutativity of multiplication and the right-distributive law. Theorem [Veblen and Wedderburn, 1907]: Non-desarguesian (seriously strange) finite geometries can be constructed from finite near-fields. Proposition: If (N, +, ) is a planar near-ring and (a b) (a = b), then (N, +, ) is a planar near-field.
(Math Sem., 4/8/04 : 10) Ferrero near-rings I Assume G = N H is a Frobenius group. Define + as the operation of N, and as below. (N, +, ) is a planar Ferrero left near-ring. F 13,4 = x : x 13 = 1 z : z 4 = 1. { 1 if x P = {1, x,..., x 12 } x k + x m = x (k+m) x m x k := m G z r (x k ) if x m z r (E) Z = {1, z, z 2, z 3 }; recall that z m (x k ) = x 5mk. Orbits under Z: {1}; {x, x 5, x 12, x 8 }; {x 2, x 10, x 11, x 3 }; {x 4, x 7, x 9, x 6 }. E := {x 2, x 4 }; Z(E) := {x 2, x 10, x 11, x 3 } {x 4, x 7, x 9, x 6 }; A := {1, x}; G := Z (A ) = {1, x, x 5, x 12, x 8 } x m 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 1 1 1 1 1 1 1 1 1 1 1 1 1 x 5 1 1 1 1 1 1 1 1 1 1 1 1 1 x 12 1 1 1 1 1 1 1 1 1 1 1 1 1 x 8 1 1 1 1 1 1 1 1 1 1 1 1 1 x 2 (1) 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 10 (z) 1 x 5 x 10 x 2 x 7 x 12 x 4 x 9 x 1 x 6 x 11 x 3 x 8 x 11 (z 2 ) 1 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 3 (z 3 ) 1 x 8 x 3 x 11 x 6 x 1 x 9 x 4 x 12 x 7 x 2 x 10 x 5 x 4 (1) 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 7 (z) 1 x 5 x 10 x 2 x 7 x 12 x 4 x 9 x 1 x 6 x 11 x 3 x 8 x 9 (z 2 ) 1 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 6 (z 3 ) 1 x 8 x 3 x 11 x 6 x 1 x 9 x 4 x 12 x 7 x 2 x 10 x 5
(Math Sem., 4/8/04 : 11) Ferrero near-rings II { 1 if x x m x k := m G z r (x k ) if x m z r (E) x m 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 1 1 1 1 1 1 1 1 1 1 1 1 1 x 5 1 1 1 1 1 1 1 1 1 1 1 1 1 x 12 1 1 1 1 1 1 1 1 1 1 1 1 1 x 8 1 1 1 1 1 1 1 1 1 1 1 1 1 x 2 (1) 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 10 (z) 1 x 5 x 10 x 2 x 7 x 12 x 4 x 9 x 1 x 6 x 11 x 3 x 8 x 11 (z 2 ) 1 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 3 (z 3 ) 1 x 8 x 3 x 11 x 6 x 1 x 9 x 4 x 12 x 7 x 2 x 10 x 5 x 4 (1) 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 7 (z) 1 x 5 x 10 x 2 x 7 x 12 x 4 x 9 x 1 x 6 x 11 x 3 x 8 x 9 (z 2 ) 1 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 6 (z 3 ) 1 x 8 x 3 x 11 x 6 x 1 x 9 x 4 x 12 x 7 x 2 x 10 x 5 (x 5 x 8 ) x 7 =1 x 7 =1 x 5 (x 8 x 7 )=x 5 1=1 x 2 (x 7 x 3 )=x 2 x 2 =x 2 (x 2 x 7 ) x 3 =x 7 x 3 =x 2 x 8 (x 2 +x 3 )=x 8 x 5 =1 x 8 x 2 +x 8 x 3 =1+1=1 (x 2 + x 3 ) x 8 = x 5 x 8 = 1 x 2 x 8 + x 3 x 8 = x 8 + x 12 = x 20
(Math Sem., 4/8/04 : 12) Ferrero pairs and nearrings An ordered pair of groups, (N, H) is a Ferrero pair if H is a fixed-point-free automorphism group of N. In particular, every Frobenius group represents forms a Ferrero pair. Theorem: Every planar near-rings is constructible as a Ferrero near-ring for some Ferrero pair. Corollary: Finite planar near-rings can be grouped into families, with each family descended from a single Frobenius group. Theorem: [Boykett, 2001] Assume that n and t are positive integers, n = p e i i. There exists a Ferrero pair (N, H) (Frobenius group N H) with N = n and H = t if and only if t divides p e i i 1 for every i. Theorem: [Boykett, 2001] (N, H) is a Ferrero pair, with N = N 1 N2, and H fixing N 1 and N 2, if and only if (N 1, H) and (N 2, H) are Ferrero pairs.
(Math Sem., 4/8/04 : 13) Combinatorial designs from near-rings I Theorem: [Clay, p. 59] Assume (N, +, ) is a finite planar near-ring. The following methods produce balanced incomplete block designs, (P, B, ). Points: Blocks BIBD? (v, b, r, k, λ) P = {n N} B = {Na + b : a, b N, a 0} Sometimes. P = {n N} B = {(N )a + b : a, b N, a 0} Always. v = N, b = v(v 1)/k, r = v 1, k = N /, λ = k 1 Example: Consider the near-ring constructed from F 13,4 = x : x 13 = 1 z : z 4 = 1. The second method should produce a BIBD on v = 13 points, with k = 3 points incident to each block. There are b = (13)(12)/3 = 52 blocks. Every point will lie on r = 12 blocks, and every pair of blocks will share λ = 2 points.
(Math Sem., 4/8/04 : 14) Combinatorial designs from near-rings II P = {n N} B = {(N )a + b : a, b N, a 0} x m 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x 1 1 1 1 1 1 1 1 1 1 1 1 1 x 5 1 1 1 1 1 1 1 1 1 1 1 1 1 x 12 1 1 1 1 1 1 1 1 1 1 1 1 1 x 8 1 1 1 1 1 1 1 1 1 1 1 1 1 x 2 (1) 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 10 (z) 1 x 5 x 10 x 2 x 7 x 12 x 4 x 9 x 1 x 6 x 11 x 3 x 8 x 11 (z 2 ) 1 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 3 (z 3 ) 1 x 8 x 3 x 11 x 6 x 1 x 9 x 4 x 12 x 7 x 2 x 10 x 5 x 4 (1) 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 7 (z) 1 x 5 x 10 x 2 x 7 x 12 x 4 x 9 x 1 x 6 x 11 x 3 x 8 x 9 (z 2 ) 1 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 x 6 (z 3 ) 1 x 8 x 3 x 11 x 6 x 1 x 9 x 4 x 12 x 7 x 2 x 10 x 5 Sample blocks: Name Expression Points Bx,1 N (x) + 1 {1, x, x 5, x 12, x 8 } Bx 2,1 N (x 2 ) + 1 {1, x 7, x 9, x 6, x 4 } Bx 3,1 N (x 3 ) + 1 {1, x 3, x 2, x 10, x 11 }
(Math Sem., 4/8/04 : 15) Combinatorial designs from near-rings III Block 1 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 B x,1 1 1 0 0 0 1 0 0 1 0 0 0 1 B x,x 1 1 1 0 0 0 1 0 0 1 0 0 0 B x,x 2 0 1 1 1 0 0 0 1 0 0 1 0 0.. B x 2,1 1 0 0 0 1 0 1 1 0 1 0 0 0 B x 2,x 0 1 0 0 0 1 0 1 1 0 1 0 0..
(Math Sem., 4/8/04 : 16) Conjectures, conclusions, etc. 1. Frobenius groups, near-rings, and combinatorial designs can all be viewed as sets with fixed-point-free automorphism groups. 2. Planar near-rings can be classified (and constructed) from Frobenius groups. 3. Frobenius complements must be cyclic or generalized quaternion groups. (a) Singer groups with dihedral quotients are rare. (b) Singer groups with cyclic quotients are strongly restricted. (c) Singer groups with quaternion quotients may be common? 4. Recent methods for constructing symmetric designs [Golemac, Vucicic] depend on identifying tactical configurations (near-bibd s) with large automorphism groups, then reducing to BIBD s with Singer groups. Can this process be formalized as identifying a Ferrero pair and corresponding Ferrero near-ring? 5. Genuinely non-abelian symmetric designs may (always?) come from geometries based on planar near-rings. 6. Communications codes produced from circular (nice planar) near-rings are particularly useful [Fuchs]. 7. There is life after Coolville.
(Math Sem., 4/8/04 : 17) References T. Boykett, Construction of Ferrero pairs of all possible orders, SIAM J. Discrete Math., Vol 14, No. 3,283-285. J. Clay, Nearrings: Geneses and Applications, Oxford Science Publications, 1992. Peter Fuchs, Gerhard Hofer, and Gunter Pilz, Codes from Planar Near Rings, IEEE Transactions in Information Theory, vol. 36, no. 3, May 1990, 647-651. P. Becker and J. Mendes, A Note on Menon-Hadamard Difference Sets in Groups of Order 4(13) 2, to be submitted. Wen-Fong Ke, On Recent Developments of Planar Nearrrings, 18th International Conference on Nearrings and Nearfields, Universitat Hamburg, September 12, 2003. www.math.uni-hamburg.de/projekte/nearring2003/abs.html A. Golemac and Tanja Vucicic, New Difference Sets in Nonabelian Groups of Order 100, Journal of Combinatorial Designs, 9, no. 6, pp. 424-434. E. Moore and H. Pollatsek, Looking for Difference Sets in Groups with Dihedral Images, Designs, Codes, and Cryptography 28, (2003), 45-50. K. Smith, Nonabelian Hadamard difference sets, Journal of combinatorial theory 70 No. 1, (1995), 144-156. Xiao Hong Wu, Difference Sets: Extraneous Multipliers and Abelianization, Doctoral dissertation, Ohio State University, 1994.