The Spectrum of Group-Based Complete Latin Squares

Ollis, M. A.; Tripp, Christopher R.

doi:10.37236/8542

Cited by 1 publication

(1 citation statement)

References 20 publications

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“…Keedwell's Conjecture [12] is that if S contains all the non-identity elements of a non-abelian group G, then a linear sequencing exists if and only if |G| ≥ 10. This known to hold for many groups, including dihedral groups [11,13], soluble groups with a single involution [3], at least one group of every odd order at which a non-abelian group exists [17] and all groups of order at most 255 [15].…”

Section: Introductionmentioning

confidence: 99%

Sequencings in Semidirect Products via the Polynomial Method

Costa¹,

Fiore²,

Ollis³

2023

Preprint

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The partial sums of a sequence x = x 1 , x 2 , . . . , x k of distinct non-identity elements of a group (G, •) are s 0 = id G and s j = j i=1 x i for 0 < j ≤ k. If the partial sums are all different then x is a linear sequencing and if the partial sums are all different when |i−j| ≤ t then x is a t-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order k of the non-identity elements of the dihedral group of order 2m has a linear sequencing when k ≤ 12 and either m > 3 is prime or every prime factor of m is larger than k!, unless s k is unavoidably the identity; that every subset of order k of a non-abelian group of order three times a prime has a linear sequencing when 5 < k ≤ 10, unless s k is unavoidably the identity; and that if the order of a group is pe then all sufficiently large subsets of the non-identity elements are t-weakly sequenceable when p > 3 is prime, e ≤ 3 and t ≤ 6.

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