Is $\langle a,b,c \mid c=aba^{-1}b \rangle$ the same as the free group $\langle a,b \rangle$?

Question

I have a group with generators $a,b$ and $c$ with a relation $c=aba^{-1}b$: $$\langle a,b,c \mid c=aba^{-1}b \rangle$$ Is this group the same as the free group $\langle a,b \rangle$? (Since $c$ is written in terms of $a$ and $b$.) I’m a bit rusty with the algebra, so I just want to make sure. Thank you.

Answer 1

Yes, the two presentations define isomorphic groups. This is simply an application of a Tietze transformation.

Answer 2

Yes, it is the free group. $c$ is redundant (in words, your presentation says: In the group generated by $a$ and $b,$ there is a word $aba^{-1}b$).