Powers of elements in Group Theory (Indices) I'm having trouble proving that the properties of powers in integers holds for elements in a group as well.
The theorem says:

Suppose $ g \in G$ and $m,n \in\Bbb Z_+$ Show that $$g^{m+n} = g^{m}*g^{n}$$

How do I prove it? I have an idea that induction should be used but what should be the base case?
Thanks!

Let's do it via induction.
Take $n=1$. Then it is obvious that $g^m* g=g^{m+1}$.
Assume that this equality it is true for $n$.
We will show that for $n+1$.
$$g^m*g^{n+1}=g^m(g^n*g)=(g^m*g^n)*g=g^{m+n}*g=g^{m+n+1}$$
Then just show that  $(g^m)^{-1}=(g^{-1})^m=g^{-m}$. for $m,n\in \mathbb{N}$.
Hence, we can deduce this equality is true for all $m,n\in \mathbb{Z}$
