# Is $\langle\ a,b\ \vert\ aba=bab,\ abab=baba\ \rangle$ a presentation of the free group on a single generator?

Is the following a presentation of the free group generated by a single element?

$$\langle\ a,b\ \vert\ aba=bab,\ abab=baba\ \rangle.$$

My thinking is the following:

$$abab = baba=b(bab)=b^2ab$$ by substituting the first relation into the second. Simplifying, we get $$a=b$$. Since these steps give equivalent statements, the above presentation is in fact $$\langle\ a,b\ \vert\ a=b\ \rangle$$, i.e., the free group on one generator.

Is this correct?

• Yes , What you have said its correct – Chirantan Chowdhury Feb 5 '17 at 18:52
• Please don't forget to accept answers using the checkmark $\checkmark$ symbol :) – Shaun Aug 18 at 20:58

One thing I would leave out is the $$b^2ab$$ step, for $$a=b$$ follows from $$abab=b(bab)$$ by cancelling $$bab$$ on the right.