Consider the following group in presentation form

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quasar987
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Something's bugging me. Consider the following group in presentation form:

[tex]<a,b|aba^{-1}b^{-1}=e,a^2=e>[/tex]

Ok, this is the presentation of a group with 2 generators that is commutative and "shrinks" any even power of one the generators to the identity. This sounds like an appropriate presentation of [itex]\mathbb{Z}/2\times \mathbb{Z}[/itex].

But the presentation can also be written in another way:

[tex]<a,b|aba^{-1}b^{-1}=e,a^2=e>=<a,b|ab=ba,a=a^{-1}>[/tex]

Now this does not resembles [itex]\mathbb{Z}/2\times \mathbb{Z}[/itex]! It says that [itex]<a,b|ab=ba,a=a^{-1}>[/itex] is a 2 generator abelian group that associates one of its generator with its inverse. Maybe I'm just tired but how does this apply to [itex]\mathbb{Z}/2\times \mathbb{Z}[/itex]??
 
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matt grime
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a=a^-1 is precisely the same as saying a^2=e. What's the problem?

the relations do not 'associate' things. They are formal relations that a and b (in this case) satisfy (and are the only such, except for those that can be deduced from them).
 
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quasar987
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I see it now. I was just tired. :approve:
 

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