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

$a^{1}=e$

$a^{2}=a*a=b$

$a^{3}=a*(a*a)=a*b=c$

$a^{4}=a*a*(a*a)=a*(a*b)=a*c=e$