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A set of permutations F of a finite transitive permutation group G ≤ Sym(Ω) is intersecting if any pair of elements of F agree on an element of Ω. We say that G has the EKR property if an intersecting set of G has size at most the order of a point stabilizer. Moreover, G has the strict-EKR property whenever G has the EKR property and any intersecting set of maximum size is a coset of a point stabilizer of G.It is known that the permutation group GL2(Fq) acting on Ωq := F 2 q \ {0} has the EKR property, but does not have the strict-EKR property since the stabilizer of a hyperplane is a maximum intersecting set. In this paper, it is proved that the Hilton-Milner type result does not hold for GL2(Fq) acting on Ωq. Precisely, it is shown that a maximal intersecting set of GL2(Fq) is of maximum size. As a result, we prove the Complete Erdős-Ko-Rado theorem for GL2(Fq).

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