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What is an Initial Segment here?: ReidemeisterSchreier Method 
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#1
Feb911, 03:43 PM

P: 662

Hi, everyone:
I am reading an article on the ReidemeisterSchreier 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 WellOrdering Principle some how? I thought we may have been considering the case where H has infinite index in G, so that we assign a wellordering 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. 


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