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I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 6.3 A Word on Free Groups ...
I have a basic question regarding the nature and character of free groups ...
Dummit and Foote's introduction to free groups reads as follows:https://www.physicsforums.com/attachments/5488In the above text, Dummit and Foote write the following:
" ... ... The basic idea of a free group $$F(S)$$ generated by a set $$S$$ is that there are no satisfied by any of the elements in $$S$$ ($$S$$ is "free"of relations.) ... ... "Dummit and Foote then show how to construct $$F(S)$$ as the set of all words (together with inverses) ... but they do not seem to prove that given that $$F(S)$$ contains all words in $$S$$ there are no relations satisfied by any of the elements in $$S$$ ...
Is the lack of a rigorous proof because the lack of any such relations is obvious ... ?
Can someone please help clarify this situation ...?
Peter
I have a basic question regarding the nature and character of free groups ...
Dummit and Foote's introduction to free groups reads as follows:https://www.physicsforums.com/attachments/5488In the above text, Dummit and Foote write the following:
" ... ... The basic idea of a free group $$F(S)$$ generated by a set $$S$$ is that there are no satisfied by any of the elements in $$S$$ ($$S$$ is "free"of relations.) ... ... "Dummit and Foote then show how to construct $$F(S)$$ as the set of all words (together with inverses) ... but they do not seem to prove that given that $$F(S)$$ contains all words in $$S$$ there are no relations satisfied by any of the elements in $$S$$ ...
Is the lack of a rigorous proof because the lack of any such relations is obvious ... ?
Can someone please help clarify this situation ...?
Peter
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