# zbMATH — the first resource for mathematics

Relative difference sets fixed by inversion. III: Cocycle theoretical approach. (English) Zbl 1159.05009
In this important paper the authors continue the application of group theory and cohomology to design theory. Specifically, let $$G$$ be a finite group. A subset $$R\subset G$$ is called a relative difference set relative to a subgroup $$N< G$$ if there exists a constant) such that (i) for every element $$g\in G\setminus N$$, there are exactly $$\lambda$$ pairs of elements $$r_1$$ and $$r_2$$ in $$R$$ such that $$g= r_1r^{-1}_2$$; and (ii) for every $$g\in N\setminus\{1\}$$, there exist no elements $$r_1$$ and $$r_2$$ in $$R$$ such that $$g= r_1r^{-1}_2$$. If $$|N|= n$$, $$|G|= mn$$, and $$|R|= k$$, the relative difference set $$R$$ is called an $$(m, n, k,\lambda)$$-relative difference set, and if $$k= m$$, the relative difference set is said to be semiregular and gives rise to a $$(m,n, k,\lambda)$$ divisible design on which the group $$G$$ acts regularly and transitively. $$N$$ is called a forbidden subgroup and is the stabilizer of a point class of the design.
The authors apply and discuss multiplicative cocycles, skew symmetric cocycles, and the connection between orthogonal cocycles and generalized Hadamard matrices. The main theorems are:
Theorem 1: Let $$G$$ be a finite group such that $$[G, G]\leq Z(G)$$, where $$[G,G]$$ is the commutator subgroup and $$Z(G)$$ is the center of $$G$$. Let $$N$$ be a subgroup of $$G$$ such that $$[G,G]\leq N\leq Z(G)$$ and $$|N|$$ divides $$|G/N|$$. The group $$G$$ contains a semiregular relative difference set $$R$$ relative to $$N$$ such that $$1\in R$$ and $$rRr= R$$ for all $$r\in R$$ if and only if $$G$$ is either an elementary abelian 2-group with $$|N|^3\leq |G|$$ and $$|G/N|$$ a perfect square, or a special $$p$$-group of exponent $$p$$ with $$|Z(G)|+ |G/Z(G)|- 1$$ conjugacy classes for some odd prime $$p$$. Furthermore, if $$|G|$$ is odd and $$R_1$$ and $$R_2$$ are two such relative difference sets in $$G$$ relative to $$N$$, then there is a homomorphism $$\delta: G\to N$$ with $$N\leq\ker(\delta)$$ such that $$R_1= \{\delta(r)r: r\in R_2\}$$.
Theorem 2: Let $$p$$ be a prime. For any positive integers $$n\leq m$$, there exists a $$(p^m, p^n,p^m,p^{m-n})$$-relative difference set $$R$$ in a group $$G$$ described in Theorem 1 if and only if $$m$$ is even and $$2n\leq m$$.
These theorems are applied to prove: For any odd prime $$p$$ and any two positive integers $$b\leq a$$, there exist $$P^{2a}\times P^{2a}$$ skew symmetric generalized Hadamard matrices over $$(\mathbb Z/p\mathbb Z)^b$$. Also, if there exists an $$(m, n, m, m/n)$$-relative difference set fixed by inversion in $$G$$ relative to a subgroup $$N$$ of $$G$$ which is contained in the center of $$G$$, then there exists an $$m\times m$$ skew symmetric generalized Hadamard matrix with entries in $$N$$.
Part II, see J. Comb. Theory, Ser. A 111, No. 2, 175–189 (2005; Zbl 1070.05016).

##### MSC:
 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
Full Text:
##### References:
 [1] Arasu, K.T.; Jungnickel, D.; Pott, A., Divisible difference sets with multiplier $$- 1$$, J. algebra, 133, 35-62, (1990) · Zbl 0706.05012 [2] Brown, K.S., Cohomology of groups, (1982), Springer New York, Heidelberg, Berlin · Zbl 0367.18012 [3] Y.Q. Chen, K.J. Horadam, W.-H. Liu, Relative difference sets fixed by inversion (II)â€”character theoretical approach, J. Combin. Theory Ser. A, to appear. · Zbl 1070.05016 [4] Y.Q. Chen, C.H. Li, Relative difference sets fixed by inversion Cayley graphs, J. Combin. Theory Ser. A, to appear. · Zbl 1066.05071 [5] Godsil, C.D.; Hensel, A.D., Distance regular covers of complete graphs, J. combin. theory ser. B, 56, 205-238, (1992) · Zbl 0771.05031 [6] Horadam, K.J.; Udaya, P., A new construction of central relative $$(p^a, p^a, p^a, 1)$$-difference sets, Designs codes cryptography, 27, 281-295, (2002) · Zbl 1027.05013 [7] Leung, K.H.; Ma, S.L., Constructions of partial difference sets and relative difference sets on $$p$$-groups, Bull. London math. soc., 22, 533-539, (1990) · Zbl 0689.05016 [8] W.-H. Liu, Central semiregular relative difference sets fixed by inversion, Ph.D. Thesis, RMIT University, 2001. [9] Ma, S.L., Reversible relative difference sets, Combinatorica, 12, 425-432, (1992) · Zbl 0769.05023 [10] MacDonald, I.D., Some $$p$$-groups of Frobenius and extra-special type, Israel J. math., 40, 350-364, (1981) · Zbl 0486.20016 [11] Perera, A.A.; Horadam, K.J., Cocyclic generalized Hadamard matrices and central relative difference sets, Designs codes cryptography, 15, 187-200, (1998) · Zbl 0919.05007 [12] J. Seberry, Some remarks on generalised Hadamard matrices and theorems of Rajkundlia on SBIBDs, Combinatorial Mathematics, vol. VI, in: Proceedings of the Sixth Australian Conference, University of New England, Armidale, 1978, pp. 154-164, Lecture Notes in Mathematics, vol. 748, Springer, Berlin, 1979. [13] Seberry, J., A construction for generalised Hadamard matrices, J. statist. plann. inference, 4, 365-368, (1980) · Zbl 0458.62068 [14] Seberry, J., Generalised Hadamard matrices and colourable designs in the construction of regular GDDs with two and three association classes, J. statist. plann. inference, 15, 237-246, (1987) · Zbl 0604.05006 [15] Suzuki, M., Group theory, (1982), Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.