Let G be a multiplicative group (not Abel's group, so it's non-commutative) and a and b are elements that belong to it. Also, a² = 1 and b³ = a^-1 * b² * a. Knowing this, show that b^5 = 1.

From a² = 1, I can see derive that a = a^-1. I have also derived that b² = ab³a and

b³ = ab²a, but I don't know if I'm on the right track. I've tried other ways too, but I always seem to end up where I started. I suppose this task isn't that hard, but I just need a little extra push to be able to solve it. I've been thinking if there's a way to state b in form b = a^n. This would solve probably everything, since a^(2n)=1.

Any help would be appreciated. Thanks in advance!