A Generalization of an Addition Theorem for Solvable Groups

Abstract

The “sets” in this paper are actually multi-sets. That is, we allow an element to occur several times in a set and distinguish between the number of elements in a set and the number of distinct elements in the set. On the few occasions when we need to avoid repetition we will use the term “ordinary set.“Definition. Let G be a group and let S a set of elements of G. An r-sum in S is an ordered subset of S of cardinality r; the result of that r-sum is the product of its elements in the designated order.Definition. If S is a set, r(x, S) denotes the number of times x appears in S and [x, S] is a set consisting of r(x, S) copies of x. An n-set or n-subset is a set consisting of n elements. Hence [x, S] is an r(x, S)-subset of S.

The Erdős–Ginzburg–Ziv theorem for dihedral groups

“…Let G be a finite non-cyclic solvable group of order n. In 1984, Yuster and Peterson [10] proved that s(G) ≤ 2n −2; in 1988, with the restriction that n ≥ 600((r − 1)!) 2 , Yuster [11] proved that s(G) ≤ 2n − r ; and in 1996, the first author [5] proved that s(G) ≤ 11 6 n − 1.…”

On the structure of n-zero-sum free sequences over cyclic groups of order n

“…The associated inverse problem of Erdős-Ginzburg-Ziv constant studies for the structure of n-zero-sum free sequences of length strictly smaller than s(G). In the 1980s, Yuster and Peterson [40] and, independently, Bialostocki and Dierker [3], first considered the structure of n-zero-sum free sequences over G of length s(G) − 1 = 2n − 2 when G is cyclic. Since then the inverse problem of s(G) in a cyclic group G has been studied by many authors.…”

On Erdős–Ginzburg–Ziv inverse theorems

“…When G is cyclic of order m, we have h(G, k) ≥ k + 1 provided m − ⌊m/4⌋ − 2 ≤ k ≤ m − 2 (see [8]); h(G, k) ≥ k + 1 provided m is prime with 1 ≤ k ≤ m − 2 (see [11]); h(G, m − 2) = m − 1 (see [1] or [15]); and h(G, m − 3) = m − 1 (see [7]). …”

“…There have been many related inverse theorems describing the structure of the sequences S in G with length |S| = m + k, 1 ≤ k ≤ m − 2, not having any m-term subsequence with zero sum. For cyclic groups of order m, the structure of S has been described by several authors: when k = m − 2, by Yuster and Peterson in [15], and by Bialostocki and Dierker in [1]; when k = m − 3, by Flores and Ordaz in [7]; when m − ⌊m/4⌋ − 2 ≤ k ≤ m − 2, by Bialostocki, Dierker, Grynkiewicz, and Lotspeich in [2] (using a related result of Gao from [8]); and when k ≥ ⌈(m − 1)/2⌉, by Chen and Savchev in [3].…”