On a classical zero-sum invariant

Abstract: Let $G$ be a nontrivial, finite abelian group. Then $ν(G)$ is the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. We study the invariant $ν(G)$, which was introduced in Zero-Sum Theory in the 1960s.

• The paper proves that ν(G) = ν₂(G) = 𝖽(G) − 1 for C₂² ⊕ C₂ₙ for all n and for C₂⁴ ⊕ C₂ₙ when n > 70 is odd, including groups where the Davenport constant exceeds its standard lower bound.
• The paper introduces the prime-index invariant νₚ(G) and local invariants ν(S), using group-algebra methods, extremal sequence classifications, and complement identities to connect missing subsequence sums with the structure of maximal minimal zero-sum sequences.
• The paper shows that these refinements constrain element orders in extremal sequences, provide a heredity principle for direct factors, and clarify why determining whether any group satisfies ν(G) = 𝖽(G) remains an open problem.

The invariant and its history

Let $G$ be a nontrivial finite abelian group. The invariant $\nu(G)$, introduced by van Emde Boas in the 1960s in his work on the Davenport constant of rank three groups (2608.19090), is defined as the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ with $|T| \ge \ell$ has the property that the set $G^{\bullet} \setminus \Sigma(T)$ of nonzero group elements not representable as subsequence sums of $T$ is contained in a proper coset $\alpha + H$ of some subgroup $H \subsetneq G$. Since every zero-sum free sequence of length $\nu(G)$0 satisfies $\nu(G)$1, one always has

$\nu(G)$2

and for cyclic groups equality holds throughout with $\nu(G)$3. Gao conjectured roughly three decades ago that $\nu(G)$4 holds for all nontrivial finite abelian groups; this remains open, and prior to this work the precise value of $\nu(G)$5 was known only for cyclic groups, $\nu(G)$6-groups, and rank two groups — all cases where additionally $\nu(G)$7. No group is known with $\nu(G)$8.

The paper makes progress on two fronts: it establishes $\nu(G)$9 for new families of groups (including groups where the Davenport constant strictly exceeds $\ell$0), and it introduces refinements — a prime-index variant $\ell$1 and local variants $\ell$2 attached to maximal zero-sum free sequences — that connect the invariant to structural questions about extremal minimal zero-sum sequences.

A refinement via index-$\ell$3 subgroups

The paper defines $\ell$4 analogously to $\ell$5 but requires that the missing subsequence sums lie in a coset of a subgroup of index exactly $\ell$6. Clearly

$\ell$7

Two exact determinations are given. For cyclic groups, the argument reduces to Olson's structure theorem for zero-sum free sequences of maximal length, forcing $\ell$8 up to translation, so the single missing element lies outside the unique index-$\ell$9 subgroup (2608.19090). For $T$0-groups, the proof is a group algebra argument over $T$1: associating to each $T$2 the scalar $T$3 defined by $T$4, where $T$5, the authors show $T$6 is a homomorphism using the identity $T$7 together with the vanishing of $T$8 at length $T$9. Any missing element $G$0 satisfies $G$1, so $G$2 with $G$3. Hence $G$4 for all finite abelian $G$5-groups.

This yields a structural corollary of independent interest: for $G$6, every element occurring in a minimal zero-sum sequence of maximal length has full order $G$7. Previously only the existence of one full-order element was known for $G$8. More generally, Proposition 3.2 shows that if $G$9 for all primes $|T| \ge \ell$0 and $|T| \ge \ell$1, then every element of every maximal-length minimal zero-sum sequence has order equal to the exponent — resolving in the affirmative, for such groups, a long-standing open conjecture on the orders of elements in extremal sequences. This illustrates how the seemingly bookkeeping invariant $|T| \ge \ell$2 controls deep structural properties of $|T| \ge \ell$3.

Groups of the form $|T| \ge \ell$4

The main result of Section 4 is that $|T| \ge \ell$5 for all $|T| \ge \ell$6. This statement had been announced by Schmid without proof. The proof rests on Schmid's classification of minimal zero-sum sequences of length $|T| \ge \ell$7 into six types (a)–(f) relative to a basis $|T| \ge \ell$8 with $|T| \ge \ell$9, $G^{\bullet} \setminus \Sigma(T)$0.

A key technical contribution is Lemma 4.3 and Corollary 4.4: after splitting type (b) into a normalized form (b*) and separating the degenerate case (d*) from (d), the authors show that the type of an extremal minimal zero-sum sequence is uniquely determined by its support partition into the sets $G^{\bullet} \setminus \Sigma(T)$1, $G^{\bullet} \setminus \Sigma(T)$2, and $G^{\bullet} \setminus \Sigma(T)$3, whose cardinalities are tabulated per type. In particular, no basis change can alter the type of a given $G^{\bullet} \setminus \Sigma(T)$4, except possibly between (d*) and (d).

The proof of the main theorem proceeds by fixing a zero-sum free $G^{\bullet} \setminus \Sigma(T)$5 with $G^{\bullet} \setminus \Sigma(T)$6 and analyzing the set of maximal sequences $G^{\bullet} \setminus \Sigma(T)$7 containing $G^{\bullet} \setminus \Sigma(T)$8, via the identity (Proposition 2.6)

$G^{\bullet} \setminus \Sigma(T)$9

which expresses the missing sums through the supports of the complements $T$0. Cases with at most two missing elements are handled directly (using that all elements of extremal sequences avoid $T$1); when $T$2 the quotient argument forces an index-2 subgroup; and the remaining types reduce to a lengthy case analysis (Lemma 4.6) built around sequences concentrated in the coset $T$3.

Groups with $T$4: the case $T$5

All previously studied groups satisfied $T$6. Theorem 5.5 breaks this pattern: for $T$7 with $T$8 odd — a family with $T$9, whose precise Davenport constant was determined only recently — the authors prove $\alpha + H$0.

The proof exploits the rigid structure of extremal sequences here: by the classification of Chen–Savchev, every $\alpha + H$1 has the form $\alpha + H$2 with $\alpha + H$3 of order $\alpha + H$4 and the remaining eight terms projecting to a distinguished 8-element subset $\alpha + H$5 satisfying Property P (all subset sums have odd cardinality). A combinatorial lemma shows that any 6-element subset $\alpha + H$6 determines $\alpha + H$7 uniquely as the complement of a subgroup of order 8. Consequently, a zero-sum free $\alpha + H$8 of length $\alpha + H$9 extending to some $H \subsetneq G$0 either has small height (in which case $H \subsetneq G$1 is unique and at most two elements are missing, handled by a linear-algebra argument over $H \subsetneq G$2) or contains $H \subsetneq G$3, in which case uniqueness of $H \subsetneq G$4 pins down the possible complements explicitly, and both candidate missing sets lie in a coset of an explicit index-2 subgroup.

The restriction $H \subsetneq G$5 odd is inherited from the underlying classification theorem and is not intrinsic to the method; whether the result extends to even $H \subsetneq G$6 or smaller odd $H \subsetneq G$7 is left open.

Local variants and their consequences

Section 6 introduces $H \subsetneq G$8 for a maximal zero-sum free sequence $H \subsetneq G$9 (one with $\nu(G)$00): the smallest threshold ensuring that subsequences $\nu(G)$01 of length at least $\nu(G)$02 have $\nu(G)$03 inside a proper coset. Two facts organize the theory:

• $\nu(G)$04 always;
• $\nu(G)$05.

Thus $\nu(G)$06 holds precisely when some extremal maximal zero-sum free sequence $\nu(G)$07 has $\nu(G)$08. Such sequences exist over prime cyclic groups, but only with $\nu(G)$09; the "standard" basis sequence $\nu(G)$10 always has $\nu(G)$11.

Theorem 6.4 gives a heredity mechanism: if $\nu(G)$12 with $\nu(G)$13 and $\nu(G)$14, then $\nu(G)$15. So verifying Gao's conjecture for a group automatically verifies it for suitable direct factors — conversely, a counterexample must propagate upward.

Two applications demonstrate the flexibility of the local viewpoint. First, the authors exhibit an explicit zero-sum free sequence $\nu(G)$16 of length $\nu(G)$17 over $\nu(G)$18, verify by exhaustive congruence analysis that it is maximal zero-sum free with $\nu(G)$19, and thereby show that $\nu(G)$20 for this family — a fact not previously recorded. Second, for $\nu(G)$21 they construct minimal zero-sum sequences of length $\nu(G)$22 with maximum multiplicity $\nu(G)$23 (all previously known examples had height $\nu(G)$24) and show that the associated near-extremal subsequences satisfy $\nu(G)$25. The latter analysis is delicate: it requires ruling out, case by case, additional missing subsequence sums using the non-degeneracy of the coefficient sequences $\nu(G)$26 and $\nu(G)$27 modulo $\nu(G)$28.

Limitations and open questions

Several caveats bound the scope of these results. The main theorems cover $\nu(G)$29 (all $\nu(G)$30) and $\nu(G)$31 ($\nu(G)$32 odd) only; the extension to $\nu(G)$33 for $\nu(G)$34, or to $\nu(G)$35 with even or small odd $\nu(G)$36, is not addressed. Gao's conjecture $\nu(G)$37 remains open in general, and the authors state plainly that in light of current knowledge of the Davenport constant it "seems to be out of reach"; they propose restricting attention to groups with $\nu(G)$38 as a realistic intermediate goal. Whether any group at all satisfies $\nu(G)$39 is unknown, though the local theory shows this would require an extremal maximal zero-sum free sequence with $\nu(G)$40. Finally, the structural corollary on element orders applies to groups $\nu(G)$41 only under the hypothesis that $\nu(G)$42 for every prime $\nu(G)$43; since $\nu(G)$44 itself is open for $\nu(G)$45 and general $\nu(G)$46, and no structural conjecture for $\nu(G)$47 exists there, the reach of this corollary is currently limited to the groups treated in Sections 3–5.

Conclusion

This paper substantially extends the class of finite abelian groups for which the classical van Emde Boas invariant is computed exactly, proving $\nu(G)$48 for $\nu(G)$49 unconditionally and for $\nu(G)$50 with $\nu(G)$51 odd — the first instances beyond groups with $\nu(G)$52. The refined invariants $\nu(G)$53 and $\nu(G)$54 are shown to be more than technical devices: $\nu(G)$55 yields the strongest known results on element orders in maximal-length minimal zero-sum sequences over elementary $\nu(G)$56-power groups, while the local variants clarify exactly what a counterexample to Gao's conjecture would require and provide a descent mechanism excluding counterexamples on direct factors. The combination of inverse-type classifications of $\nu(G)$57 with the complement identity for missing subsequence sums emerges as an effective general strategy, one whose applicability to higher-rank groups remains the central open question raised by this work.