- #1

- 4,807

- 32

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]??

[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]??

Last edited: