[Paper Review] Terraces for Small Groups

M. A. Ollis|arXiv (Cornell University)|Mar 4, 2016

Advanced Mathematical Theories and Applications11 references3 citations

TL;DR

This paper uses heuristic and backtracking algorithms in GAP to investigate terraces and sequencings in small groups up to order 511. It confirms Bailey’s Conjecture (all non-cyclic elementary abelian 2-groups are the only non-terrace groups) up to order 511, except possibly at 256 and 384, and verifies Keedwell’s Conjecture (all non-abelian groups of order ≥10 are sequenceable) up to order 255, with additional verification for specific groups like A₆, S₆, and PSL(2,q) and PGL(2,q) for small q. It also reports the first directed T₂-terrace for a non-cyclic odd-order group (order 21), constructs narcissistic and half-and-half terraces for non-abelian groups of order 27 and 39, and provides exhaustive counts of essentially different terraces and directed terraces for all groups of order ≤15.

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 $3 \leq q_1 \leq 11$ and $3 \leq q_2 \leq 8$. 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.

Motivation & Objective

• To test Bailey’s Conjecture that only non-cyclic elementary abelian 2-groups lack terraces, up to group order 511.
• To verify Keedwell’s Conjecture that all non-abelian groups of order at least 10 are sequenceable, up to order 255 and for specific families of groups.
• To construct special terraces—such as directed T₂-terraces, half-and-half terraces, and narcissistic terraces—for non-abelian groups of small odd and even orders.
• To compute the total number of essentially different terraces and directed terraces for all groups of order at most 15.
• To provide computational evidence and constructions that support the existence of complete Latin squares at new odd orders, including 63.

Proposed method

• Employed heuristic hill-climbing algorithms in GAP to search for directed terraces and sequencings in small groups.
• Used backtracking algorithms to construct special terraces such as T₂-terraces, half-and-half terraces, and narcissistic terraces.
• Implemented group-theoretic transformations, including left-multiplication to normalize terraces to basic form for equivalence checking.
• Applied the concept of 'essentially different' terraces by testing group automorphisms after normalization.
• Validated results against known values and corrected prior errors in the literature, such as t(ℤ₅) and t(ℤ₃²).
• Provided code and data at a public website for reproducibility and further research.

Experimental results

Research questions

• RQ1 Does Bailey’s Conjecture—that only non-cyclic elementary abelian 2-groups fail to have a terrace—hold for all groups of order up to 511?
• RQ2 Are all non-abelian groups of order at least 10 sequenceable, as per Keedwell’s Conjecture, up to order 255 and for specific families like PSL(2,q) and PGL(2,q)?
• RQ3 Can directed T₂-terraces be constructed for non-cyclic groups of odd order, and if so, what are the first such examples?
• RQ4 Do non-abelian groups of odd order admit narcissistic or half-and-half terraces, and can such terraces be used to generate infinite families?
• RQ5 What are the exact counts of essentially different terraces and directed terraces for all groups of order at most 15?

Key findings

• Bailey’s Conjecture holds up to order 511, with possible exceptions only at orders 256 and 384.
• Keedwell’s Conjecture is verified up to order 255 and for A₆, S₆, PSL(2,q₁) with 3≤q₁≤11, and PGL(2,q₂) with 3≤q₂≤8.
• The first directed T₂-terrace for a non-cyclic group of odd order (order 21) was constructed, yielding the first known Roman-2 square of odd order.
• Both non-abelian groups of order 27 possess narcissistic and directed half-and-half terraces, and G₃₉,₁ has a directed half-and-half terrace.
• The paper reports 31 new orders at which complete Latin squares are now known to exist, with the smallest such order being 63.
• Exhaustive enumeration confirms the number of essentially different terraces and directed terraces for all groups of order ≤15, correcting prior errors in the literature.