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An explicit terrace for every non-abelian group of order 256 and 384

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Zenodo2026-09-16 更新2026-10-01 收录
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Explicit terraces for all 76,224 non-abelian groups of order 256 (56,070 groups) and order 384 (20,154 groups), given as 76,325 witness records; 101 groups of order 384 carry two terraces. A terrace of a group G of order n is an ordering a_1, ..., a_n of all its elements, starting at the identity, whose differences b_i = a_i^-1 a_{i+1} contain every self-inverse element exactly once and every other element together with its inverse exactly twice in total. The deposit is built for checking rather than for trust in how the terraces were found. sh run_all.sh runs four checks using only the Python 3 standard library, with no GAP, network or installation: the files match their checksums; every record is a terrace of a genuine group of the stated order (the verifier first proves the generators give a regular permutation group of that order); nine deliberately wrong negative controls are all rejected; and every non-abelian id of each order is present, counted against NrSmallGroups(n) minus the abelian groups. A fifth check, scripts/check_ids.g, needs GAP and confirms that each record's group is the SmallGroup its id names. Both checks were run on these exact files and their unedited logs ship here (verification_log.txt, ids_check_log.txt), each headed by the SHA-256 of every file it read: 76,325 / 76,325 witnesses verified, and 76,325 / 76,325 ids confirmed with 0 mismatches. Orders 256 and 384 were the cases left open by Theorem 32(vi) of M. A. Ollis, Sequenceable Groups and Related Topics (Electronic Journal of Combinatorics, Dynamic Survey DS10, Version 3, 2025), which covers every other non-abelian order up to 511. Combined with that survey, every non-abelian group of order at most 511 is terraced.

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2026-09-16
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