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Small unordered Davenport constant of the order-125 Heisenberg group

For the exponent-five Heisenberg group of order 125, determine the small unordered Davenport constant: the greatest length of a sequence with no nonempty subsequence whose terms can be ordered to multiply to the identity. The target value is twelve, requiring a length-twelve example and an upper bound for every thirteen-term sequence.

Why it matters and what counts as progress

Why it matters: It determines the first stated open prime case of the Heisenberg small Davenport-constant prediction, with sequence order treated as unrestricted.

What counts as progress: Match a product-one-free construction with an upper bound for all thirteen-term sequences; the unordered subsequence invariant is the target.

How the problem developed

  1. 4 Nov 2023 · Special case [1]

    Naveen K. Godara and Siddhartha Sarkar Proved that the small Davenport constant of the exponent-3 non-abelian group of order 27 is 6, providing the lower-prime case in this Heisenberg-group line of work.

  2. 15 Jul 2026 · Exact result [2]

    Patrick White Proved d(H_125)=12; the upper bound reduces to a finite spread bound checked by exhaustive C and Python searches, with auxiliary lemmas machine-checked.

AI contribution

Sources

  1. Godara and Sarkar, A note on a Conjecture of Gao and Zhuang for groups of order 27
  2. Patrick White, The small Davenport constant of the Heisenberg group of order 125
  3. The small Davenport constant of the Heisenberg group of order 125
  4. Heisenberg Davenport verification code
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