quasar987
Feb14-07, 12:56 AM
Something's bugging me. Consider the following group in presentation form:
<a,b|aba^{-1}b^{-1}=e,a^2=e>
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 \mathbb{Z}/2\times \mathbb{Z}.
But the presentation can also be written in another way:
<a,b|aba^{-1}b^{-1}=e,a^2=e>=<a,b|ab=ba,a=a^{-1}>
Now this does not resembles \mathbb{Z}/2\times \mathbb{Z}! It says that <a,b|ab=ba,a=a^{-1}> 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 \mathbb{Z}/2\times \mathbb{Z}??
<a,b|aba^{-1}b^{-1}=e,a^2=e>
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 \mathbb{Z}/2\times \mathbb{Z}.
But the presentation can also be written in another way:
<a,b|aba^{-1}b^{-1}=e,a^2=e>=<a,b|ab=ba,a=a^{-1}>
Now this does not resembles \mathbb{Z}/2\times \mathbb{Z}! It says that <a,b|ab=ba,a=a^{-1}> 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 \mathbb{Z}/2\times \mathbb{Z}??