 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 BIRS Workshop Lecture Videos /
 Comparing system of sets of lengths over finite abelian...
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Comparing system of sets of lengths over finite abelian groups Schmid, Wolfgang
Description
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 zerosum sequence. We denote by $\mathcal{B}(G)$ the set of all zerosum sequences over $G$. This is a submonoid of the monoid of all sequences over $G$. We say that a zerosum sequence is a minimal zerosum sequence if it cannot be decomposed into two nonempty zerosum 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 factorizationlength of $S$ if there are minimal zerosum 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 factorizationlength 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 $G'$.
Item Metadata
Title 
Comparing system of sets of lengths over finite abelian groups

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20191112T09:46

Description 
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 zerosum sequence. We denote by $\mathcal{B}(G)$ the set of all zerosum sequences over $G$.
This is a submonoid of the monoid of all sequences over $G$. We say that a zerosum sequence is a minimal zerosum sequence if it cannot be decomposed into two nonempty zerosum 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 factorizationlength of $S$ if there are minimal zerosum 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 factorizationlength 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 $G'$.

Extent 
44.0 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: Universite Paris 8

Series  
Date Available 
20200511

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0390442

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International