bibkey: gordon1961sequences
authors: Basil Gordon
year: 1961
title: Sequences in groups with distinct partial products
doi: 10.2140/pjm.1961.11.1309
url: https://msp.org/pjm/1961/11-4/p14.xhtml
claim: Theorem 1 constructs a sequencing of a nontrivial finite abelian group with a unique involution; its mixed-radix prefixes give each nonidentity directed difference once.
strata_touched:
  - D5/S3/FiniteGroups/GordonCarry
  - D5/S3/FiniteGroups/GordonProduct
  - D5/S3/FiniteGroups/GordonFullSet
license: citation-only
triage: anchor

Sequences in groups with distinct partial products

Basil Gordon, Pacific Journal of Mathematics 11 (1961), 1309–1313, Theorem 1, proof (ii), pp. 1309–1311.

Gordon starts the ordering with the identity and includes all nonempty partial products. In additive notation, deleting that first zero gives a word containing each nonzero group element once. Its empty prefix and all its nonempty prefixes are pairwise distinct. The nonzero-total hypothesis excludes the identity-only group, which is a degenerate exception to the literal printed equivalence.

The construction interleaves the negatives of mixed-radix coordinates with their coordinatewise translates by one. Odd coordinate moduli make doubling invertible. The first nonzero digit of a predecessor carry can therefore be recovered from the directed difference. A complete borrow through zero gives exactly zero. No coprimality between odd radices, including repeated factors, is required; Chinese remaindering is used only between the even prime-power factor and the product of the odd radices.

The theorem concerns the full nonzero group. Deleting labels changes prefix sums and does not settle the corresponding arbitrary-subset problem. The mathematics and construction are published standard results; no originality is claimed. The primary article is cited without redistributing its PDF.

Verified locator

DOI: https://doi.org/10.2140/pjm.1961.11.1309

Publisher: https://msp.org/pjm/1961/11-4/p14.xhtml