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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.

##### MSC:
 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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##### References:
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