Relative difference sets in semidirect products with an amalgamated subgroup
✍ Scribed by John C. Galati; Alain C. LeBel
- Publisher
- John Wiley and Sons
- Year
- 2005
- Tongue
- English
- Weight
- 140 KB
- Volume
- 13
- Category
- Article
- ISSN
- 1063-8539
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✦ Synopsis
Abstract
We call a group G with subgroups G~1~, G~2~ such that G = G~1~G~2~ and both N = G~1~ ∩ G~2~ and G~1~ are normal in G a semidirect product with amalgamated subgroup N. We show that if G~l~ is a group with N~l~ ⊲ G~l~ containing a relative $({m_l},n,{m_l},{{m_l}\over{n}})$‐difference set relative to N~l~ for l = 1,2, and if there exists a “compatible coupling” from (G~2~, N~2~) to (G~1~, N~1~), a notion introduced in the paper, then for any i,j ∈ ℕ there exists at least one semidirect product with amalgamated subgroup N ≅ N~1~ ≅ N~2~ containing a relative $(m_1^im_2^j,n, m_1^im_2^j, {{m_1^im_2^j}\over {n}})$‐difference set. We say “at least one” to emphasize that the proof is via recursive construction and that different groups may be obtained depending on the choices made at different stages of the recursion. A special case of this result shows that if K is any finite group containing a normal relative ${(m,n,m,}{{m}\over {n}})$‐difference set, then there exists, for each i ∈ ℕ, at least one semidirect product with amalgamated subgroup N containing a relative $({m^i},n,{m^i},{{m_i}\over {n}})$‐difference set. These results suggest that the class of semidirect products with an amalgamated subgroup provides a rich source of new (non‐abelian) semiregular relative difference sets. © 2004 Wiley Periodicals, Inc.