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# Group presentation terminology confusion watch

1. I have a very rudimentary question about terminology. We have defined a relation in an alphabet X as a nonempty reduced word in X. This makes sense when we move to defining the presentation of a group as a form where R is a set of relations.

However, we also "emphasise" that a word () is a relation by writing , but I don't see how that emphasises anything! I understand writing relations in that form is useful when it comes to working with the presentation of the group (as it gives us relations (in the colloquial sense) which define the group), but I don't see the justification for writing a reduced word "=1".

What am I missing?
2. I think it's just to emphasise a declaration. All just notation really. "F(X) with relations r1, r2, r3" is as well understood as "F(X) with relations r1=1, r2=1, r3=1". Though in the latter you could remove the word "relations". You couldn't really do the same with the former.
3. Also, it gives you liberty to write things like "ab = ba" (which we understand as commutativity of those elements) rather than "aba-1b-1" (which is slightly less clear). As in lots of areas of maths, it's huge notational fudging for occasional convenience or aesthetics, but as long as you understand it that's fine.

So <a, b | ab = ba> = "the group generated by a and b where ab = ba" is very obviously Z^2. On the other hand, <a, b | aba-1b-1> looks nowhere near as intuitive.

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Updated: February 12, 2010
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