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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.)
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##### References:
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