The large Davenport constant I: Groups with a cyclic, index 2 subgroup

By Alfred Geroldinger, David J. Grynkiewicz. 2012

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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 constant d(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 constant D(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.

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/95f6c146a228a59259556c77730dbb99f9e5f4a5

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In Discrete Mathematics and Combinatorics

  1. K. Cziszter
  2. Mátyás Domokos
  3. Alfred Geroldinger
  4. Wolfgang A. Schmid

In Electrical and Electronic Engineering

  1. K. Cziszter
  2. Mátyás Domokos
  3. Alfred Geroldinger
  4. Wolfgang A. Schmid

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