Consider the following group in presentation form

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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 $\mathbb{Z}/2\times \mathbb{Z}$??

Last edited:

matt grime