Show that $\langle a,b | ababa \rangle $ is a presentation of $\mathbb{Z}$.

Question

I have to solve the following exercise:

$\langle a,b | ababa \rangle $ is a presentation of $\mathbb{Z}$.

Hint: Let $t =ab$

How can you I show such a thing? Help would be very appreciated.

Answer 1

Following the hint, we have

$$t^2 a = 1 \implies a = t^{-2}$$

and

$$t = ab \implies b = t^3$$

In particular, this shows that $\langle a, b | ababa\rangle$ is cyclic - so it suffices to check that $\mathbb{Z}$ actually satisfies this presentation. This is immediate from an appropriate choice of $a$ and $b$.

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Note the general pattern of the proof: We first find a "bound" on the size of the group, given the presentation - in this case, the role of the bound is played by the conclusion that the group is cyclic, which is quite limiting. Then we find a sufficiently large group that satisfies the presentation.