Product-one Sequences Research Articles

On product-one sequences over subsets of groups

Let G be a group and G_0 subseteq G be a subset. A sequence over G_0 means a finite sequence of terms from G_0, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. We study algebraic and arithmetic properties of monoids of product-one sequences over finite subsets of G and over the whole group G, with a special emphasis on the infinite dihedral group.

On product-one sequences with congruence conditions over non-abelian groups

Let G be a finite group. For a positive integer d, let sdN(G) denote the smallest integer ℓ such that every sequence S over G of length |S|≥ℓ has a nonempty product-one subsequence T with |T|≡0 (mod d). In this paper, we mainly study this invariant for dihedral groups D2n and metacyclic groups Cp⋉sCq.

On product-one sequences over dihedral groups

Let [Formula: see text] be a finite group. A sequence over [Formula: see text] means a finite sequence of terms from [Formula: see text], where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over [Formula: see text] (with the concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This paper provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

On the algebraic and arithmetic structure of the monoid of product-one sequences

Let G be a finite group. A finite unordered sequence S=g1 ∙⋯ ∙gℓ of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals 1G, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid ℱ(G) with basis G, and we study the submonoid ℬ(G)⊂ℱ(G) of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if G is abelian. In case of abelian groups, ℬ(G) is a well-studied object. In the present paper we focus on nonabelian groups, and we study the class semigroup and the arithmetic of ℬ(G).

On an inverse problem of Erdős, Kleitman, and Lemke

Let (G,1G) be a finite group and let S=g1⋅…⋅gℓ be a nonempty sequence over G. We say S is a tiny product-one sequence if its terms can be ordered such that their product equals 1G and ∑i=1ℓ1ord(gi)≤1. Let ti(G) be the smallest integer t such that every sequence S over G with |S|≥t has a tiny product-one subsequence. The direct problem is to obtain the exact value of ti(G), while the inverse problem is to characterize the structure of long sequences over G which have no tiny product-one subsequence. In this paper, we consider the inverse problem for cyclic groups and we also study both direct and inverse problems for dihedral groups and dicyclic groups.

On Erdős-Ginzburg-ZIV inverse theorems for dihedral and dicyclic groups

Let G be a finite group and exp(G) = lcm{ord(g) ∣ g ∈ G}. A finite unordered sequence of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of G. We denote by s(G)(or E(G) respectively) the smallest integer l such that every sequence of length at least l has a product-one subsequence of length exp(G)(or ∣G∣ respectively). In this paper, we provide the exact values of s(G) and E(G) for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length s(G) — 1 (or E(G) — 1 respectively) having no product-one subsequence of length exp(G)(or ∣G∣ respectively).

On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is, a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.

On the algebraic and arithmetic structure of the monoid of product-one sequences II

Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The monoid $\mathcal B (G)$ of all product-one sequences over $G$ is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if $G$ is abelian (equivalently, $\mathcal B (G)$ is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if $|G'| \le 2$ if and only if $\mathcal B (G)$ is seminormal, and we study sets of lengths in $\mathcal B (G)$.

Erdős–Ginzburg–Ziv theorem and Noether number for Cm⋉φCmn

Let G be a multiplicative finite group and S=a1⋅…⋅ak a sequence over G. We call S a product-one sequence if 1=∏i=1kaτ(i) holds for some permutation τ of {1,…,k}. The small Davenport constant d(G) is the maximal length of a product-one free sequence over G. For a subset L⊂N, let sL(G) denote the smallest l∈N0∪{∞} such that every sequence S over G of length |S|≥l has a product-one subsequence T of length |T|∈L. Denote e(G)=max⁡{ord(g):g∈G}. Some classical product-one (zero-sum) invariants including D(G):=sN(G) (when G is abelian), E(G):=s{|G|}(G), s(G):=s{e(G)}(G), η(G):=s[1,e(G)](G) and sdN(G) (d∈N) have received a lot of studies. The Noether number β(G) which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let G≅Cm⋉φCmn, in this paper, we prove thatE(G)=d(G)+|G|=m2n+m+mn−2 and β(G)=d(G)+1=m+mn−1. We also prove that smnN(G)=m+2mn−2 and provide the upper bounds of η(G), s(G). Moreover, if G is a non-cyclic nilpotent group and p is the smallest prime divisor of |G|, we prove that β(G)≤|G|p+p−1 except if p=2 and G is a dicyclic group, in which case β(G)=12|G|+2.

The large Davenport constant II: General upper bounds

Let G be a finite group written multiplicatively. By a sequence over G, we mean a finite sequence of terms from G which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of G. The small Davenport constantd(G) is the maximal integer ℓ such that there is a sequence over G of length ℓ which has no nontrivial, product-one subsequence. The large Davenport constantD(G) is the maximal length of a minimal product-one sequence—this is a product-one sequence which cannot be partitioned into two nontrivial, product-one subsequences. The goal of this paper is to present several upper bounds for D(G), including the following: D(G)≤{d(G)+2|G′|−1,where G′=[G,G]≤G is the commutator subgroup;34|G|,if G is neither cyclic nor dihedral of order 2n with n odd;2p|G|,if G is noncyclic, where p is the smallest prime divisor of |G|;p2+2p−2p3|G|,if G is a non-abelian p-group.As a main step in the proof of these bounds, we will also show that D(G)=2q when G is a non-abelian group of order |G|=pq with p and q distinct primes such that p∣q−1.

The large Davenport constant I: Groups with a cyclic, index [formula omitted] subgroup

Let G be a finite group written multiplicatively. By a sequence over G, we mean a finite sequence of terms from G which is unordered, repetition of terms allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of G. The small Davenport constantd(G) is the maximal integer ℓ such that there is a sequence over G of length ℓ which has no nontrivial, product-one subsequence. The large Davenport constantD(G) is the maximal length of a minimal product-one sequence—this is a product-one sequence which cannot be factored into two nontrivial, product-one subsequences. It is easily observed that d(G)+1≤D(G), and if G is abelian, then equality holds. However, for non-abelian groups, these constants can differ significantly. Suppose G has a cyclic, index 2 subgroup. Then an old result of Olson and White (dating back to 1977) implies that d(G)=12|G| if G is non-cyclic, and d(G)=|G|−1 if G is cyclic. In this paper, we determine the large Davenport constant of such groups, showing that D(G)=d(G)+|G′|, where G′=[G,G]≤G is the commutator subgroup of G.