Consider the following group in presentation form

[quasar987]

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

 

[matt grime]

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).

 

[quasar987]

I see it now. I was just tired.