View source for Schreier system

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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:

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

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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.S. Massey,   "Algebraic topology: an introduction" , Springer  (1977)</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)  pp. 93</TD></TR></table>