# Schreier system

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:

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

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 [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=33003