The structure of maximal zero-sum free sequences

Inverse zero-sum problems II

“…By a very recent result (see [11], and [9] for an earlier partial result) it is known that to establish Property B for C 2 m for each m ∈ N, it suffices to establish it for C 2 p for each prime p. Moreover, Property B is known to hold for C 2 m for m ≤ 28 (see [2] and [9] for m ≤ 7). For further recent results towards establishing Property B see [16,12,2].…”

Inverse zero-sum problems III

“…(a) For every n ∈ [2,10], the group C n ⊕ C n has Property B: for n ≤ 6 this may be found in [5,Proposition 4.2]; the cases n ∈ {8, 9, 10} (and more) are settled in [1].…”

“…The next three paragraphs detail the three major types of swaps that we will use. If U, V ∈ A 1 are distinct (thus U = W i and V = W j for some i and j distinct), then we may exchange any subsequence X|U for a subsequence Y |V with |Y | = |X| (if U = W 0 , then X must additionally lie within W (1) 0 , and likewise for V ) and the resulting product decomposition W ′ will still lie in Ω 0 , equal to W except that the blocks U and V of W have been replaced by the blocks U ′ := X −1 U Y and…”

“…Suppose W 0 ∈ C 0 . Then we may choose f 2 such that σ(W 0 ) = f 1 + f 2 , and if Ω and y|W j with ǫ ′ (x, y) / ∈ {1, n}, whence Lemmas 5.3.3 and 3.1.2 imply that supp(ψ(W (1) 0 )) = {γ, β} (say) with β − γ = ±F j (else CLAIM A follows), where…”

Zero-sum free sets with small sum-set

“…In fact, our main motivation for carrying out these computations was that we needed this corollary ( [1], Lemma 9) to deduce some general results on zero-sum free sequences in Z p × Z p and Proposition 1.1(4) was not strong enough to deal with small primes.…”