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What is an Initial Segment here?: Reidemeister-Schreier Method

  1. Feb 9, 2011 #1
    Hi, everyone:

    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.
  2. jcsd
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