I'm trying to make up an example of a quotient of a free group to check if I understand quotients properly. I do for the usual cases but I've not seen free groups before. So here I go:
Let $F = \langle x, y \rangle$ and let $N$ be the group that is generated by the element $x^5 y^{-2} = 1$, i.e. the relation $x^5 = y^2$.
Is it right then that
F/N = \{ x, x^2, \dots , x^5, \dots, x^6y^3x^6, \dots, x^6y^{-3}, \dots\} / N = \{ x, x^2, \dots , x^5, \dots, x^6y^3x^6, \dots, xy^{-1}, \dots\}, i.e. all the elements in $N$ get reduced to $1$, even if they're sandwiched between two expressions?
Thanks for your help!