Although Rajesh's answer is best way to approach this problem , yet these is another approach.
We can observe given group is cyclic group , where 'a' is generator , and every cyclic group is abelian too. So fill the missing entries on the bases of commutative property of abelian groups
$a^{1}=e$
$a^{2}=a*a=b$
$a^{3}=a*(a*a)=a*b=c$
$a^{4}=a*a*(a*a)=a*(a*b)=a*c=e$