On terraces for abelian groups

Abstract

We use heuristic algorithms to find terraces for small groups. We show that Bailey's Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and 384. We also show that Keedwell's Conjecture (that all non-abelian groups of order at least 10 are sequenceable) holds up to order 255, and for the groups A 6 , S 6 , PSL(2, q 1 ) and PGL(2, q 2 ) where q 1 and q 2 are prime powers with 3 ≤ q 1 ≤ 11 and 3 ≤ q 2 ≤ 8. A sequencing for a group of a given order implies the existence of a complete latin square at that order. We show that there is a sequenceable group for each odd order up to 555 at which there is a non-abelian group. This gives 31 new orders at which complete latin squares are now known to exist, the smallest of which is 63. In addition, we consider terraces with some special properties, including constructing a directed T 2 -terrace for the non-abelian group of order 21 and hence a Roman-2 square of order 21 (the first known such square of odd order). Finally we report the total number terraces and directed terraces for groups of order at most 15.MSC: 20D60, 20B15.

On invertible terraces for non‐abelian groups

“…It is known that the elementary abelian groups of order at least 4 do not have terraces [6]. All other abelian groups, except possibly those of the form A × Z for A an abelian group of order coprime to 6 and k ≥ 1, do have terraces [6,11]. Hence there is a conjugate-quasicomplete Latin square of every order.…”

Constructions for Terraces and R-Sequencings, Including a Proof That Bailey's Conjecture Holds for Abelian Groups

“…The main R-sequencing result is Theorem 4.1 that lets us show that many groups with elementary abelian Sylow 2subgroups of order 8 are R-sequenceable. We then employ a trick introduced in [16] that…”

“…Theorem 4.4 [17]. To prove that all abelian groups other than the noncyclic elementary abelian 2-groups are terraced it is sufficient to show that groups of the form Z 3 2 × Z p are terraced for all primes p ≥ 7.…”

“…The next result on this topic for abelian groups was that Z n × Z 2 is terraced for n = 2 [2]. More recently, a series of papers by the current authors [16,17,19] has shown that the only possible abelian exceptions to Bailey's Conjecture are those whose Sylow 2-subgroup is Z 3 2 . In Section 4, we show that such groups (other than Z 3 2 itself) are indeed terraced.…”

“…The circular sequence b is called a rotational sequencing, or R-sequencing and the circular sequence a is called a directed rotational terrace or directed R-terrace. Various naturally equivalent definitions and alternative naming conventions are used in the literature [1,10,13,16,20].…”