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**Relative difference sets in semidirect products with an amalgamated subgroup.**
*(English)*
Zbl 1067.05013

In recent years there has been growing interest in the construction of non-abelian examples of semiregular relative difference sets. The authors give a new construction method by using what they call {semidirect product with amalgamated subgroups}, that is, a group \(G\) with subgroups \(G_1\) and \(G_2\) such that \(G=G_1G_2\) and both \(N=G_1 \cap G_2\) and \(G_1\) are normal in \(G\). Given relative difference sets with parameters \((m_l,n,m_l,\frac{m_l}{n})\) in two groups \(G_l\) (\(l=1,2\)), relative to normal subgroups \(N_l\) of \(G_l\), the authors give a sufficient condition (the existence of a {compatible coupling}) for the existence of a relative difference set with parameters \((m_1^im_2^j, n,m_1^im_2^j, \frac{m_1^im_2^j}{n})\) in some semidirect product with amalgamated subgroup \(N \cong N_1 \cong N_2\) for all positive integers \(i\) and \(j\). They also discuss several examples for this construction.

Reviewer: Dieter Jungnickel (Augsburg)

### MSC:

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

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\textit{J. C. Galati} and \textit{A. C. LeBel}, J. Comb. Des. 13, No. 3, 211--221 (2005; Zbl 1067.05013)

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### References:

[1] | Presentations of Groups, 2nd edition, London Mathematical Society Student Texts, Vol. 15, Cambridge University Press, Cambridge, 1997. |

[2] | Galati, J Combin Designs 11 pp 307– (2003) |

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[4] | and Relative difference sets fixed by inversion (III)?cocycle theoretical approach, Discrete Math, to appear. · Zbl 1159.05009 |

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[6] | Finite geometry and character theory, Springer-Verlag, Berlin, 1995. · Zbl 0818.05001 |

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