Properties of Abelian Groups I'm trying to prove that if $G$ is an Abelian group under $\cdot$, $\forall a,b \in G. \forall z \in \mathbb{Z}. (a \cdot b)^n = a^n \cdot b^n.$ I was originally considering doing this problem using an AFSOC, but I realized that originally assuming that $(a \cdot b)^n \neq a^n \cdot b^n$ would be rather difficult to contradict with the given properties of an Abelian group. Thus, I considered inducting in two ways on $n$, first going through all the positives and then going through all the negatives in the integers. However, I'm worried about what I do with the case of $n = 0.$ I understand that in algebraic terms $a^0$ is an abbrevation for ``$a \cdot a \cdot ... \cdot a$ with 0 many $a$'s'', but I am confused as to what this represents in the Abelian group $G.$ Perhaps I need to prove some properties of $a^0$ for all $a \in G$?

$G$ is an abelian group, so let $a,b\in G$ be given, and fix $n$. Then $$ (ab)^n=(ab)(ab)(ab)\cdots(ab).$$ Because $G$ is abelian $$ (ab)(ab)(ab)\cdots(ab)=(aa\cdots a)(bb\cdots b)=a^nb^n.$$ This follows naturally from the definition of commutativity.