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Difference between revisions of "Schreier system"

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A non-empty subset of a [[Free group|free group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834001.png" /> with set of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834002.png" />, satisfying the following condition. Let an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834003.png" /> of the Schreier system be represented as a reduced word in the generators of the group:
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A non-empty subset of a [[Free group|free group]] $F$ with set of generators $S$, satisfying the following condition. Let an element $g\neq 1$ of the Schreier system be represented as a reduced word in the generators of the group:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834004.png" /></td> </tr></table>
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$$g=S_1^{n_1}\ldots S_k^{n_k},$$
  
 
and let
 
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834005.png" /></td> </tr></table>
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$$g'=\begin{cases}gS_k,&n_k<0,\\gS_k^{-1},&n_k>0.\end{cases}$$
  
It is required then, that the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834006.png" /> should also belong to this system (the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834007.png" /> can be considered as the reduced word obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083400/s0834008.png" /> by deleting its last letter). The element 1 belongs to every Schreier system.
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It is required then, that the element $g'$ should also belong to this system (the element $g'$ can be considered as the reduced word obtained from $g$ by deleting its last letter). The element 1 belongs to every Schreier system.
  
 
Introduced by O. Schreier in the 1920s, see [[#References|[1]]].
 
Introduced by O. Schreier in the 1920s, see [[#References|[1]]].

Revision as of 11:45, 19 August 2014

A non-empty subset of a free group $F$ with set of generators $S$, satisfying the following condition. Let an element $g\neq 1$ of the Schreier system be represented as a reduced word in the generators of the group:

$$g=S_1^{n_1}\ldots S_k^{n_k},$$

and let

$$g'=\begin{cases}gS_k,&n_k<0,\\gS_k^{-1},&n_k>0.\end{cases}$$

It is required then, that the element $g'$ should also belong to this system (the element $g'$ can be considered as the reduced word obtained from $g$ by deleting its last letter). The element 1 belongs to every Schreier system.

Introduced by O. Schreier in the 1920s, see [1].

References

[1] W.S. Massey, "Algebraic topology: an introduction" , Springer (1977)


Comments

Of particular interest are Schreier systems which are systems of representations of the cosets of a subgroup. Cf. [a1] for some uses of Schreier systems, such as a proof of the Nielsen–Schreier theorem that subgroups of free groups are free.

References

[a1] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) pp. 93
How to Cite This Entry:
Schreier system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schreier_system&oldid=33000
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article