Relative difference sets fixed by inversion. III: Cocycle theoretical approach.

*(English)*Zbl 1159.05009In 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).

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).

Reviewer: Spencer P. Hurd (Charleston)

##### MSC:

05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |

##### Keywords:

relative difference set; orthogonal cocycle; special \(p\)-group; generalized Hadamard matrix
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\textit{Y. Q. Chen} et al., Discrete Math. 308, No. 13, 2764--2775 (2008; Zbl 1159.05009)

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