All difference family structures arise from groups

Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additiv...

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Дата:	2009
Автор:	Boykett, T.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2009
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	All difference family structures arise from groups / T. Boykett // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ.

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spelling	irk-123456789-1533802019-06-15T01:26:31Z All difference family structures arise from groups Boykett, T. Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference. We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups. This result will be of interest to those using nonstandard algebras as bases for defining 2-designs. 2009 Article All difference family structures arise from groups / T. Boykett // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/153380 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Planar nearrings have been used to define classes of 2-designs since Ferrero's work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay's work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference. We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups. This result will be of interest to those using nonstandard algebras as bases for defining 2-designs.
format	Article
author	Boykett, T.
spellingShingle	Boykett, T. All difference family structures arise from groups Algebra and Discrete Mathematics
author_facet	Boykett, T.
author_sort	Boykett, T.
title	All difference family structures arise from groups
title_short	All difference family structures arise from groups
title_full	All difference family structures arise from groups
title_fullStr	All difference family structures arise from groups
title_full_unstemmed	All difference family structures arise from groups
title_sort	all difference family structures arise from groups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/153380
citation_txt	All difference family structures arise from groups / T. Boykett // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 20–31. — Бібліогр.: 17 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT boykettt alldifferencefamilystructuresarisefromgroups
first_indexed	2025-07-14T04:36:20Z
last_indexed	2025-07-14T04:36:20Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2009). pp. 20 – 31 c© Journal “Algebra and Discrete Mathematics” All difference family structures arise from groups Tim Boykett Communicated by G. Pilz Dedicated to Cheryl Praeger on the occasion of her 60th Birthday Abstract. Planar nearrings have been used to define classes of 2-designs since Ferrero’s work in 1970. These 2-designs are a class of difference families. Recent work from Pianta has generalised Ferrero and Clay’s work with planar nearrings to investigate planar nearrings with nonassociative additive structure. Thus we are led to the question of nonassociative difference families. Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalized difference family constructions could be made, using the standard basis of translation and difference. We determine minimal axioms for a difference family structure to give a 2-design. Using these minimal axioms we show that we obtain quasigroups. These quasigroups are shown to be isotopic to groups and the derived 2-designs from the nonassociative difference family are identical to the 2-designs from the isotopic groups. Thus all difference families arise from groups. This result will be of interest to those using nonstandard alge- bras as bases for defining 2-designs. 1. Introduction An interesting application of nearring theory has been in the generation of designs. The theory of planar nearrings and Ferrero Pairs was introduced by G. Ferrero [7] and heavily investigated by a group including J. Clay and W–F. Ke [2, 6, 10, 11]. The core of this work is the use of orbits of fixed- point-free automorphisms on groups in order to define a difference family Key words and phrases: Algebras, 2-design, difference family, nonassociative. Jo u rn al A lg eb ra D is cr et e M at h .T. Boykett 21 structure. Recently S. Pianta [14] has investigated the generalization of planar nearrings with nonassociative addition. A natural question then follows: can the design ideas from Ferrero and Clay be generalized to Pianta’s expanded class? Rather than investigate this from the nearring side, we decided to see how to generalize difference family structures to nonassociative additions, as this should prove to be useful for other investigators looking at generalizing other algebraic structures from which difference families can be defined. Difference families are used to construct many 2–designs. The class of 2–designs that arise from difference families can be easily characterized [1, 5] as designs with a group of automorphisms that is strictly one- transitive on points. There have been many uses of them in order to find 2–designs, in addition to the references above we note the work of Furino[8] and Buratti [3, 4]. Difference families are based upon groups, usually additively written. The essential operations are the difference operation (subtraction in the group) and translation (the mapping ob- tained by adding a fixed element). The main requirement for a difference family is that the difference between two arbitrary elements must remain invariant under translation. In this paper we look at minimal structures of translation and difference which allow a generalized form of difference family construction. We will see that although we can generalize consid- erably, all such structures arise in a clear fashion from groups. Quasigroups and loops are generalizations of groups that do not re- quire the operation to be associative. A quasigroup is a (2)–algebra (S, +) such that for all a, b ∈ S all equations a + x = b y + a = b have unique solutions for x and y. The Cayley tables of such algebras form Latin squares. There are many special cases of such algebras. In particular a loop is a quasigroup with a two–sided identity e ∈ S, i.e. a+e = e+a = a ∀a ∈ S. A group is an associative loop. See for instance [13] for details about loops and quasigroups. Quasigroups can be obtained by twisting a group in some way. A simple example is to take an additive group (G, +) and to use the sub- traction operation to obtain a quasigroup (G,−) that is in general not associative. For another example, given a field K, let k ∈ K, k 6= 0, 1 be arbitrary but fixed. Define a ∗ b = ka + (1 − k)b. Then (K, ∗) is a quasigroup, in general nonassociative. There exists a more general form of equivalence between quasigroups, or more general algebras. Two groupoids (S, +) and (T, ∗) are isotopic if there exist bijections α, β, γ : S → T such that for all a, b ∈ S α(a + b) = β(a) ∗ γ(b). An isomorphism is an isotopism with all bijections identical. Jo u rn al A lg eb ra D is cr et e M at h .22 All difference family structures arise from groups The isotope of a quasigroup is a quasigroup, so the class of quasigroups is closed under isotopism. We note that the class of groups is not closed under isotopism. In fact many proper quasigroups of interest are isotopic to groups, for instance those examples above. An (2, 2)–algebra (N, +, ∗) such that (N, +) and (N, ∗) are quasigroups is a biquasigroup. In the following we will first look at difference families and determine what properties are needed in order to be useful for such a construction. We demonstrate that such a structure is equivalent to a class of biquasi- groups with constants. We then look at this class and see that they are all simply obtained from groups. The difference family structures come directly from a group. Our main results are Proposition 16 which gives an explicit construction of all such biquasigroups and Proposition 17 demonstrating that the difference families are identical. In general we are interested in finite structures. However, the results here also apply for infinite structures. Note also that many of these results are folklore in the quasigroup community. However, because this article is aimed an combinatorialists investigating general algebraic structures for designs, we include proofs using elementary methods. 2. Difference families A (set) 2–design is a pair (V,B), where V is a set (of points), B is a set of subsets of V all of size k (called blocks) and for all pairs a, b ∈ V, a 6= b, \|{B ∈ B : a, b ∈ B}\| = λ for some constants k and λ. The number 2 in the name refers to the pairs of elements a, b. There are many variations on this definition, see e.g. [1, 5] for details. Given a (2)–algebra (N, +) and a set of subsets B of N , define the development of B in N , dev(B) to be the collection {B + n : B ∈ B, n ∈ N}, possibly containing duplicates. The set development is the collection with no duplicates. Given a group (N, +), not necessarily abelian, and a set B = {Bi : i = 1, . . . , s} of subsets of N , called base blocks, such that • all Bi have the same size • for all B, C ∈ B, n ∈ N , B + n = C ⇔ B = C and n = 0 • there exists some λ such that for all nonzero d ∈ N , \|{(B, a, b) : B ∈ B, a, b ∈ B, a − b = d}\| = λ Then B forms a difference family (DF) in the group (N, +). Theorem 1 (See e.g. [5, 1]). Let B be a difference family on a group (N, +). Then (N, devB) is a set 2-design. Jo u rn al A lg eb ra D is cr et e M at h .T. Boykett 23 In the proof of this result, we can see that the requirement that (N, +) be a group is too strong. We use only the translation and difference operations. Thus it would seem that this construction can be generalized to be based upon other structures. The following result does this. Theorem 2. Let N be a set with binary operation − (difference) and unary operations ti ∈ T (translations) and B a set of subsets of N such that • for all a, b ∈ N there is a unique ti such that tia = b. • for all a, b ∈ N , the equation a − x = b has a unique solution. • a − b = tia − tib for all a, b ∈ N , for all ti. • there exists some λ such that for all α, β ∈ N , α 6= β, \|∆(α−β)\| = λ where ∆(d) = {(B, a, b) : B ∈ B, a, b ∈ B, a 6= b, a − b = d}. • there exists some integer k such that \|tiB\| = k for all ti for all B ∈ B. • tiB = tjC for B, C ∈ B implies i = j and B = C. Then (N, devB) with devB = {tiB : B ∈ B} is a set 2-design. Proof. All the blocks in devB have size k by construction. They are all distinct by the final requirement. We need only show that the number of blocks on a pair of points is constant. Let a 6= b ∈ N . We show that \|∆(a − b)\| = \|{tiB : ti ∈ T, B ∈ B, a, b ∈ tiB}\|. There are exactly λ triples (B, α, β) in ∆(a−b) such that α, β ∈ B, α− β = a− b. For each (B, α, β) in ∆(a− b) there is a unique ti such that tiα = a. We know a − b = α − β = tiα − tiβ = a − tiβ so by the unique solution property of difference, b = tiβ. Thus a, b ∈ tiB, so we have a mapping Θ from ∆(a − b) into {tiB : ti ∈ T, B ∈ B, a, b ∈ tiB}. This map Θ is injective by the final condition. We now show that Θ is surjective. Let a, b ∈ tiB. Then there exist some α, β ∈ B such that a = tiα, b = tiβ, α − β = tiα − tiβ = a − b so (B, α, β) ∈ ∆(a−b), and tiB is in the image of Θ. Thus Θ is a bijection and we are done. We now have a generalized form of difference family. In the next sections we will investigate the algebraic properties underlying this result. Jo u rn al A lg eb ra D is cr et e M at h .24 All difference family structures arise from groups 3. Algebraic properties Let us investigate the algebraic properties of the results above. For this section, let N , ti and − be as defined in Theorem 2 above. Definition 3. Fix a0 ∈ N . For all b ∈ N , let tb ∈ T such that tba0 = b. This is unique. Define x + y := tyx. Note that the first condition of Theorem 2 above then states that the equation a + x = b has a unique solution. Lemma 4. There exists a unique o ∈ N that is a right additive zero on N . Proof. Fix a ∈ N . By the unique solution property of translations, there is some o ∈ T such that a + o = a. Then for all b ∈ N , a − b = (a + o) − (b + o) = a − (b + o), so by the unique solution property of differences, b + o = b and o is a right additive zero on N . If some p ∈ N were also a right additive zero, then a + o = a + p = a so by the unique solution property o = p and we see that o is unique. Theorem 5. (N,−) and (N, +) are both quasigroups. Proof. We know a− x = b has a unique solution. Suppose x− a = b has two solutions, x1 6= x2. There is some y such that x1 + y = x2. Then x1 − a = (x1 + y) − (a + y) = x2 − (a + y) = b = x2 − a Then x2−x = b has solutions a and a+y for x. Thus a+y = a so y = o, x1 = x2 and x − a = b has at most one solution. Now we have to show that at least one solution to the equation x−a = b exists. Let c ∈ N be arbitrary but fixed, then c − x = b has a unique solution x1. By the above argument the equation x − x1 = b has the unique solution c. We also know that the equation x1 + x = a has a unique solution x2. Then b = c − x1 = (c + x2) − (x1 + x2) = (c + x2) − a shows that c + x2 is a solution to x − a = b, so (N,−) is a quasigroup. We now turn our attention to the addition operation. As noted above, a + x = b has a unique solution. Suppose x + a = b has two solutions x1, x2. Then x1 − x2 = (x1 + a)− (x2 + a) = b− b. There is some unique k such that b+k = x1. Then x1−x2 = b− b = (b+k)− (b+k) = x1−x1 so x1 = x2 by quasigroup property of (N,−), so x + a = b has at most one solution. Jo u rn al A lg eb ra D is cr et e M at h .T. Boykett 25 Let d be arbitrary but fixed. Then x − d = b − (d + a) has a unique solution x1 by the quasigroup property of (N,−). Then x1 − d = (x1 + a) − (d + a) = b − (d + a) so due to the unique solution property of x−(d+a) = x1−d we know that x1+a = b, so x1 is a solution to x+a = b and (N, +) is a quasigroup. We have shown that the structure used in Theorem 2 can be seen as a set with two operations that form quasigroups. We formalize this, as it is clear that from such a pair of quasigroups we can form the translations used in Theorem 2. Definition 6. A difference family biquasigroup (DFBQ) (N, +,−) is a (2, 2)–algebra that is a biquasigroup such that the identity a − b = (a + c) − (b + c) is satisfied. Lemma 7. There is a constant e ∈ N such that e = a − a for all a. Proof. Fix some a ∈ N . Define e := a − a. For all b ∈ N , there exists some c such that a + c = b. Thus b − b = (a + c) − (a + c) = a − a = e and the second statement is proved. We may write a DFBQ as (N, +,−, o, e) where o is the right additive identity and e = a − a for all a. 4. In general In this section we examine the structure of a general DFBQ. We will use these results in the next section to demonstrate that a general DFBQ is isotopic to a group and that the resulting designs are identical. Definition 8. A collection B and a DFBQ (N, +,−) such that: • there exists an integer k such that \|B\| = k for all B ∈ B • there exists some λ such that for all α, β ∈ N , α 6= β, \|∆(α−β)\| = λ where ∆(d) = {(B, a, b) : B ∈ B, a, b ∈ B, a 6= b, a − b = d}. • B + b = C + c for B, C ∈ B, b, c ∈ N implies B = C and b = c. is called a quasigroup difference family (QDF) It is clear that such a QDF will give a 2-design using the same methods as Theorem 2. Jo u rn al A lg eb ra D is cr et e M at h .26 All difference family structures arise from groups Proposition 9. Let (N, +,−, o, e) be a DFBQ. Let ē be such that e+ ē = o, then define φ : x 7→ x + ē and α : x 7→ x − e. Define the operations a ⊕ b = φ−1(φa + φb) (1) a ⊖ b = α−1(a − b) (2) Then (N,⊕,⊖, e, e) is a DFBQ with a ⊖ e = a for all a ∈ N . Proof. (N,⊕) and (N,⊖) are quasigroups by isotopism. Note that φa − φb = a − b by the DFBQ property, so φ−1a − φ−1b = a − b. Then (a ⊕ c) ⊖ (b ⊕ c) = α−1((a ⊕ c) − (b ⊕ c)) (3) = α−1((φa + φc) − (φb + φc)) (4) = α−1(φa − φb) (5) = α−1(a − b) (6) = a ⊖ b (7) so we have the DFBQ property. The constants are both e since a ⊕ e = φ−1(φa + φe) = φ−1(φa + o) = a and a ⊖ a = α−1(a − a) = α−1(e) = e. The second claim is seen by αa = a−e thus a⊖e = α−1(a−e) = a. Note that a⊖ b = φa⊖ φb. This is important for the next result, the converse. Proposition 10. Let (N,⊕,⊖, e, e) be a DFBQ with a ⊖ e = a for all a ∈ N . Let φ be a permutation of N such that a ⊖ b = φa ⊖ φb, α a permutation of N such that αe = e. Define a + b = φ(φ−1a ⊕ φ−1b) (8) a − b = α(φ−1a ⊖ φ−1b) (9) Let o := φe. Then (N, +,−, o, e) is a DFBQ, αa = a − e, φa = a + ē where e + ē = o. Proof. (N, +) and (N,−) are quasigroups by isotopism. The DFBQ prop- erty is seen by (a + c) − (b + c) = α(φ−1φ(φ−1a ⊕ φ−1c) ⊖ φ−1φ(φ−1b ⊕ φ−1c)) (10) = α((φ−1a ⊕ φ−1c) ⊖ (φ−1b ⊕ φ−1c)) (11) = α(φ−1a ⊖ φ−1b) (12) = a − b (13) The constants are given by a + o = φ(φ−1a ⊕ φ−1o) = φ(φ−1a ⊕ e) = a and a − a = α(a ⊖ a) = αe = e. Jo u rn al A lg eb ra D is cr et e M at h .T. Boykett 27 Then α(a) = α(a⊖ e) = α(φ−1a⊖ φ−1e) = a− e. Let ē be such that e + ē = o. Then e + ē = φ(φ−1e ⊕ φ−1ē) = o = φe so φ−1e ⊕ φ−1ē = e. Then a = a ⊖ e = φ−1a ⊖ φ−1e (14) = (φ−1a ⊕ φ−1ē) ⊖ (φ−1e ⊕ φ−1ē) (15) = (φ−1a ⊕ φ−1ē) ⊖ e (16) = φ−1a ⊕ φ−1ē (17) = φ−1(a + ē) (18) Thus φa = a + ē and we are done. Definition 11. A quasigroup (Q, ◦) is a Ward quasigroup if (a ◦ c) ◦ (b ◦ c) = a ◦ b for all a, b, c ∈ Q. Ward first investigated these structure in [17], Furstenberg referred to the equation above in [9]. Johnson and Vojtechovsky give further historical details in Section 2 of [16], where we also find the following. Theorem 12 ([16] Theorem 2.1). Let (Q, ◦) be a Ward quasigroup. Then there exists a unique element e ∈ Q such that for all x ∈ Q, x ◦ x = e. Define x̄ = e ◦ x and x ∗ y = x ◦ ȳ for all x, y ∈ Q. Then (Q, ∗,̄ ) is a group, and x ◦ y = x ∗ ȳ. Proposition 13. Let (N, +,−, e, e) be a DFBQ with a− e = a for all a. Then it is isotopic to a group. Proof. Let I be the permutation of N such that a + Ia = e. Then a − b = (a + Ib) − (b + Ib) = (a + Ib) − e = a + Ib. (19) Thus (a − b) − (c − b) = (a + Ib) − (c + Ib) = a − c so (N,−) is a Ward quasigroup. Thus there is a group (N, ∗, ·−1) with a − b = a ∗ b−1 and a + b = a ∗ (I−1b)−1 by equation (19). The converse of this result holds too. The proof is simple calculation. Lemma 14. Let (N, ∗, 1) be a group, I a permutation of N fixing 1. Define a + b = a ∗ (Ib)−1 (20) a − b = a ∗ b−1 (21) Then (N, +,−, 1, 1) is a DF biquasigroup with a − 1 = a for all a. Jo u rn al A lg eb ra D is cr et e M at h .28 All difference family structures arise from groups Thus we obtain information on the form of the map φ in Proposition 10. We know the form of the operations from Proposition 13 so we can make some explicit statements about the structure. Corollary 15. Let (N,⊕,⊖, e, e) and φ be as for Proposition 10. Let the operation ∗ be as from Proposition 13. Then there exists some k ∈ N such that the map φ is of the form φ(a) = a ∗ k. Conversely, given (N,⊕,⊖, e, e) as in Proposition 10 and a binary operation ∗ such that (N, ∗) is a group as in Proposition 13, select any el- ement k ∈ N . Then φ(a) := a∗k satisfies the requirements of Proposition 10. Proof. By Proposition 13 we know that a⊖ b = a ∗ b−1. Since φa⊖ φb = a⊖b we have φa∗(φb)−1 = a∗b−1. Let b = 1 and we obtain φa∗(φ1)−1 = a so φa = a ∗ φ1. Letting k := φ1 we are done. The converse is seen by taking any element k ∈ N . Define φa := a∗k. Then φa ⊖ φb = (a ∗ k) ∗ (b ∗ k)−1 = a ∗ b−1 = a ⊖ b so we are done. 5. General explicit descriptions In this section, we will look at explicit descriptions of DFBQs and QDFs. Using the results above, we know the structure of all DFBQs. Proposition 16. Let (N, ∗, 1) be a group. Let α, β be permutations of N , α1 = 1. Define a + b = a ∗ βb (22) o = β−1(1) (23) a − b = α(a ∗ b−1) (24) Then (N, +,−, o, 1) is a DFBQ and all DFBQs are of this form. Proof. The forward direction is a calculation and is clear. Let (N, +,−, o, e) be a DFBQ. We demonstrate that there exists a group structure (N, ∗,−1, 1) and permutations α, β of N as above. By Proposition 9 there exist φ and α such that defining a ⊕ b := φ−1(φa + φb) (25) a ⊖ b := α−1(a − b) (26) we obtain (N,⊕,⊖, e, e) is a DFBQ with a ⊖ e = a. By Proposition 13 there exists some group (N, ∗,−1, 1) such that e = 1, a⊖ b = a ∗ b−1 and a ⊕ b = a ∗ (I−1(b))−1. Thus a + b = φ(φ−1a ∗ (I−1(φ−1b))−1) (27) a − b = α(a ∗ b−1) (28) Jo u rn al A lg eb ra D is cr et e M at h .T. Boykett 29 By Corollary 15 we know that φx = x ∗ k, φ−1x = x ∗ k−1. Thus a + b = ((a ∗ k−1) ∗ (I−1(b ∗ k−1))−1) ∗ k (29) = a ∗ k−1 ∗ (I−1(b ∗ k−1))−1 ∗ k (30) = a ∗ β(b) (31) where β(x) = k−1 ∗ (I−1(x ∗ k−1))−1 ∗ k is a permutation of N . Since a + β−1(1) = a ∗ β(β−1(1)) = a ∗ 1 = a we know β−1(1) is the unique right identity, so o = β−1(1). The permutation α fixes e which is seen to be 1 and we are done. This final result shows that all difference family structures are in fact group structures. Proposition 17. The quasigroup development and the group develop- ment of a difference family are identical. Proof. Suppose we have a QDF B on a DFBQ (N, +,−, o, e). By Prop 16 above, we know that there is a group operation ∗ and some permutation of N such that a + b = a ∗ β(b). Thus if B is a subset of N , dev+B = {B+n : n ∈ N} = {B∗β(n) : n ∈ N} = {B∗n : n ∈ N} = dev∗B so we obtain exactly the same set of sets. Thus dev+B = dev∗B and we are done. 6. Conclusion It would be desirable to generalize the definition of a difference family so as to use more general structures to derive designs using this formalism. With simple and reasonable requirements for our difference family struc- tures, we have shown that we obtain a biquasigroup algebra and that such algebraic structures must be isotopic to groups. It is also seen that the resulting designs are identical, the central result in this paper. Thus we see that there is no need to bring in complex algebraic struc- tures in order to obtain new designs through the difference family method. We also see that should such a construction work in some algebraic struc- ture, we can make some simple manipulations and obtain a group struc- ture, enhancing our knowledge of the algebraic structure. Questions remain open as to whether the requirements that we posit are all necessary. It may be reasonable to use a simpler structure for the difference and translation operations, however the postulates we use seem to be minimal. Jo u rn al A lg eb ra D is cr et e M at h .30 All difference family structures arise from groups Several applications can be seen here. For instance, planar nearrings have been shown to possess a difference family structure. Questions about nonassociative planar nearrings have been raised [14], and it might be ap- propriate to use these results to deduce structure about the nonassociative nearrings that could be so defined. The investigation of neardomains and K–loops [12] suggests that there are some strange and interesting proper- ties when we drop the finiteness and associativity restriction. In particu- lar there may be connections between the generalization of nearfields to neardomains and the generalization to planar nearrings and Ferrero pairs [6, 15], which may be connected to the construction of nonassociative difference families. 7. Acknowledgements This work has been supported by grants P15691 and P19463 of the Austrian National Science Foundation (Fonds zur Förderung der wis- senschaftlichen Forschung). I would also like to thank Petr Vojtechovsky for many ideas and support with loops and quasigroup theory. References [1] Thomas Beth, Dieter Jungnickel, and Hanfried Lenz. Design theory. Vol. I. Cam- bridge University Press, Cambridge, second edition, 1999. [2] Gerhard Betsch and James R. Clay. Block designs from Frobenius groups and planar near–rings. In Proceedings of the Conference on Finite Groups, pages 473–502. Academic Press, 1976. [3] M. Buratti. Pairwise balanced designs from finite fields. Discrete Mathematics, 208/209:103–117, 1999. [4] M. Buratti. A description of any regular or 1–rotational design by difference methods. In Combinatorics 2000, 2000. [5] Charles J. Colbourn and Jeffrey H. Dinitz, editors. Handbook of combinatorial designs. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2007. [6] James R. Clay. Nearrings: Geneses and applications. Oxford University Press, 1992. [7] Giovanni Ferrero. Stems planari e BIB–desegni. Riv. Mat. Univ. Parma, pages 79–96, 1970. [8] S. Furino. Difference families from rings. Discrete Mathematics, 97:177–190, 1991. [9] H. Furstenberg. The inverse operation in groups. PAMS, 6(6):991–997, 1955. [10] W-F Ke. Structures of circular planar nearrings. PhD thesis, University of Ari- zona, 1992. [11] W-F Ke and H Keichle. Automorphisms of certain design groups. J. Algebra, 167:488–500, 1994. [12] Hubert Kiechle. Theory of K-loops. Number 1778 in Lecture Notes in Mathemat- ics. Springer Verlag, 2002. Jo u rn al A lg eb ra D is cr et e M at h .T. Boykett 31 [13] Hala O. Pflugfelder. Quasigroups and Loops: Introduction. Number 7 in Sigma Series in Pure Mathematics. Heldermann, 1990. [14] Silvia Pianta. Loop nearrings. In Hubert Kiechle, Alexander Kreuzer, and Momme Johs Thomsen, editors, Nearrings and Nearfields; Proceedings of the Conference on Nearrings and Nearfields, Hamburg, Germany 2003, pages 57–68, 2005. [15] Günter Pilz. Near–rings. Number 23 in Mathematics Studies. North–Holland, 1983. [16] Petr Vojtechovsky and Kenneth W. Johnson. Right division in groups, Dedekind- Frobenius group matrices, and Ward quasigroups. Abh. Math. Sem. Univ. Ham- burg, 2006. [17] M. Ward. Postulates for the inverse operations in a group. TAMS, 32(3):520–526, 1930. Contact information T. Boykett Department of Algebra, Johannes–Kepler Universität Linz and Time’s Up Research Department E-Mail: tim@timesup.org Received by the editors: 18.03.2009 and in final form 18.03.2009.

All difference family structures arise from groups

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