The focus of this study is to classify flag-transitive 2-designs. We have come to the conclusion that if D is a nontrivial 2-design having block size 5 and G is a two-dimensional projective special linear group which acts flag-transitively onDwith q ≢ 1 (mod 4), then D is a 2-(11, 5, 2) design, a 2-(11, 5, 12) design, a 2-(q + 1, 5, 2(q − 1)) design with q ≡ 3 (mod 4) or a 2-(q + 1, 5, (q − 1)/3) design with q 2f (where f > 2 is an even).
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