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I am reading an article on the Reidemeister-Schreier

method for finding a presentation of a subgroup H of

a group G, given a presentation for G , in which this

statement is made:

A Schreier transversal of a subgroup H of F, free with

basis X, is a subset T of F such that for distinct

t in T, the cosets Ht are distinct, and the union of

the Ht is F, and such that ...

** every initial segment of an element of T itself

belongs to T **

Now, I understand that the cosets of H in G

partition G, and we select a subset T of G so that

Ht=/Ht' for t,t' in T, and \/Ht =G , but I have no

idea of what an initial segment would mean in this

context; are we assuming there is some sort of ordering

in T; maybe inherited from G ,or are we using

Well-Ordering Principle some how?

I thought we may have been considering the case where H

has infinite index in G, so that we assign a well-ordering in G

so that we can use Choice to select the least element g

representing the class Hg (i.e., all g_i in G with Hg_i=Hg ), but

I am not too clear on this.

Thanks for Any Ideas.