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Comparing system of sets of lengths over finite abelian groups

Banff International Research Station for Mathematical Innovation and Discovery

For $(G,+,0) $ a finite abelian group and $S= g_1 \dots g_k $ a sequence over $G$, we denote by $\sigma(S)$ the sum of all terms of $S$. We call $|S|=k$ the length of the sequence. If the sum of $S$ is $0$, we say that $S$ is a zero-sum sequence. We denote by $\mathcal{B}(G)$ the set of all zero-sum sequences over $G$. This is a submonoid of the monoid of all sequences over $G$. We say that a zero-sum sequence is a minimal zero-sum sequence if it cannot be decomposed into two non-empty zero-sum subsequences. In other words, this means that it is an irreducible element in $\mathcal{B}(G)$. For $S \in \mathcal{B}(G)$ we say that $\ell$ is a factorization-length of $S$ if there are minimal zero-sum sequences $A_1, \dots , A_{\ell}$ over $G$ such that $S = A_1 \dots A_l$. We denote the set of all $\ell$ that are a factorization-length of $S$ by $\mathsf{L}(S)$. The set $\mathcal{L}(G) = \{\mathsf{L}(G) \colon B \in \mathcal{B}(G) \}$ is called the system of sets of lengths of $G$. Obvioulsy isomorphic groups have the same system of sets of lengths. The questions arises whether the converse is true, that is, whether $\mathcal{L}(G) = \mathcal{L}(G')$ implies that $G$ and $G'$ are isomorphic. The standing conjecture is that except for two couples of groups this is indeed true. We survey partial progress towards this problem. Relatedly, if $G \subset G'$ is a subgroup, then $\mathcal{L}(G) \subset \mathcal{L}(G')$. We also present recent results, obtained together with A. Geroldinger, on the problem of establishing (and ruling out) such inclusions in cases where $G$ is not a subgroup of $G'$.

Mathematics; Combinatorics; Number theory; Discrete mathematics

video/mp4

Author affiliation: Universite Paris 8

2020-05-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74391

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/