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Constructions for Terraces and R‐Sequencings, Including a Proof That Bailey's Conjecture Holds for Abelian Groups
Author(s) -
Ollis M. A.,
Willmott Devin T.
Publication year - 2015
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21396
Subject(s) - sylow theorems , mathematics , abelian group , order (exchange) , locally finite group , conjecture , combinatorics , elementary abelian group , torsion subgroup , rank of an abelian group , omega and agemo subgroup , pure mathematics , group (periodic table) , finite group , physics , finance , quantum mechanics , economics
Every abelian group of even order with a noncyclic Sylow 2‐subgroup is known to be R‐sequenceable except possibly when the Sylow 2‐subgroup has order 8. We construct an R‐sequencing for many groups with elementary abelian Sylow 2‐subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2‐groups have terraces. For odd orders it is known that abelian groups are R‐sequenceable except possibly those with noncyclic Sylow 3‐subgroups. We show how the theory of narcissistic terraces can be exploited to find R‐sequencings for many such groups, including infinitely many groups with each possible of Sylow 3‐subgroup type of exponent at most 3 12 and all groups whose Sylow 3‐subgroups are of the formZ 3 ρ × Z 9 ρ × Z 27 σorZ 3 ρ × Z 9 ρ + 1 × Z 27 σ .