The group $S_3$ of permutation of $\{1,2,3\}$ acts on the three-dimensional vector space over the finite field $\Bbb F_3$ of three elements, by permuting the vectors in basis $\{e_1,e_2,e_3\}$ by $$\sigma\cdot e_i=e_{\sigma(i)}, \quad i=1,2,3$$ for all $\sigma \in S_3$. The cardinality of the set of the vectors fixed under the above action is:
a) 0
b) 3
c) 9
d) 27
Attempt: If $\sigma$ is identity permutation then $\sigma\cdot e_i=e_{\sigma(i)}=e_i$ so all the basis elements will be fixed. For $\sigma$ non identity we can proceed further but that will be long and time taking, So I am looking for general formula and approach to tackle this type of problem.