To prove that $a^m.a^n=a^{m+n}$ 
In a group $G$, I want to prove this theorem:
$$\forall a\in G,\; a^m.a^n=a^{m+n},\;m,n\in \Bbb Z.$$

I am thinking that only associative law is sufficient to prove this.please give some suggestions or hint to prove this.thanks in advance

Write out what each exponent means. We have
$$\begin{align}
a^ma^n&=\underbrace{a\times \dots \times a}_{m\text{ times}}\times\underbrace{a\times \dots \times a}_{n\text{ times}} \\
&=\underbrace{a\times \dots \times a}_{m+n\text{ times}} \\
&=a^{m+n}.
\end{align}$$
Yes, associativity is thus sufficient.
