If you were given only a and b, then there is a way to prove this, but since you are given c as well, I am not sure.

Given: abab = e where e is the unit.

b(abab) = b

b(abab)b^(-1) = bb^(-1)

(baba)(bb^(-1)) = e

baba = e

P.S. Questions about groups should be posted to the University Algebra forum.