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In the area of [[mathematics]] called [[combinatorial group theory]], the '''Schreier coset graph''' is a [[graph (mathematics)|graph]] associated to a [[group (mathematics)|group]] ''G'', a [[Generating set of a group|generating set]] { ''x''<sub>''i''</sub> : ''i'' in ''I'' }, and a [[subgroup]] ''H'' ≤ ''G''.
 
 
==Description==
The [[vertex (graph theory)|vertices of the graph]] are the right [[coset]]s ''Hg'' = { ''hg'' : ''h'' in ''H'' } for ''g'' in ''G''.
 
The [[edge (graph theory)|edges of the graph]] are of the form (''Hg'',''Hgx<sub>i</sub>'').
 
The [[Cayley graph]] of the group ''G'' with { ''x''<sub>''i''</sub> : ''i'' in ''I'' } is the Schreier coset graph for ''H'' = { 1<sub>G</sub> },{{harv|Gross|Tucker|1987|p=73}}. 
 
A [[spanning tree]] of a Schreier coset graph corresponds to a Schreier transversal, as in [[Schreier's subgroup lemma]], {{harv|Conder|2003}}.
 
The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of [[groupoid]]s. A subgroup ''H'' of a group ''G'' determines a covering morphism of groupoids <math> p: K \rightarrow G </math> and if ''X'' is a generating set for ''G'' then its inverse image under ''p'' is the Schreier graph of ''(G,X)''.
 
==Name==
The graph is named after [[Otto Schreier]]. 
==Applications==
The graph is useful to understand [[coset enumeration]] and the [[Todd–Coxeter algorithm]].
 
Coset graphs can be used to form large [[permutation representation]]s of groups and were used by [[Graham Higman]] to show that the [[alternating group]]s of large enough degree are [[Hurwitz group]]s, {{harv|Conder|2003}}.
 
Every [[vertex-transitive graph]] is a coset graph.
 
== References ==
* {{Citation | last1=Conder | first1=Marston |authorlink=Marston Conder| title=Groups St. Andrews 2001 in Oxford. Vol. I | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | id={{MathSciNet | id = 2051519}} | year=2003 | volume=304 | chapter=Group actions on graphs, maps and surfaces with maximum symmetry | pages=63–91}}
* {{Citation | last1=Gross | first1=Jonathan L. | last2=Tucker | first2=Thomas W. | title=Topological graph theory | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley-Interscience Series in Discrete Mathematics and Optimization | isbn=978-0-471-04926-5 | id={{MathSciNet | id = 898434}} | year=1987}}
* [http://arxiv.org/abs/0911.2915 Schreier graphs of the Basilica group Authors: Daniele D'Angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda]
[[Category:Combinatorial group theory]]
 
* Philip J. Higgins, Categoriues and Groupoids, van Nostrand, New York, Lecture Notes, 1971, [http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html Republished as TAC Reprint, 2005]
 
 
{{algebra-stub}}

Latest revision as of 09:55, 9 December 2013

In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated to a group G, a generating set { xi : i in I }, and a subgroup HG.


Description

The vertices of the graph are the right cosets Hg = { hg : h in H } for g in G.

The edges of the graph are of the form (Hg,Hgxi).

The Cayley graph of the group G with { xi : i in I } is the Schreier coset graph for H = { 1G },Template:Harv.

A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma, Template:Harv.

The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids. A subgroup H of a group G determines a covering morphism of groupoids and if X is a generating set for G then its inverse image under p is the Schreier graph of (G,X).

Name

The graph is named after Otto Schreier.

Applications

The graph is useful to understand coset enumeration and the Todd–Coxeter algorithm.

Coset graphs can be used to form large permutation representations of groups and were used by Graham Higman to show that the alternating groups of large enough degree are Hurwitz groups, Template:Harv.

Every vertex-transitive graph is a coset graph.

References

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

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  • Schreier graphs of the Basilica group Authors: Daniele D'Angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda


Template:Algebra-stub