Let $x \in G$ and let $a,b \in \mathbb{Z}^+$. Prove that $x^{a+b}=x^a*x^b$.

Question

I think this requires induction. Fix a and induction on b. For the base case i have: Let b=1 then x^(a+1)=x^a+x^b. Im not sure how to proceed to the induction hypothesis.

Answer 1

You may also use induction on $a+b$: $$ x^{a+b+1} = x^{a+b} x = (x^a x^b) x = x^a (x^b x) = x^a x^{b+1} $$ The first equality is by definition. The second equality is by induction. So is the last one.