Flag-Transitive 2-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mfenced open="(" close=")" separators="|"><a:mrow><a:mi>v</a:mi><a:mo>,</a:mo><a:mn>5</a:mn><a:mo>,</a:mo><a:mi>λ</a:mi></a:mrow></a:mfenced></a:math> Designs Admitting a Two-Dimensional Projective Group

The focus of this study is to classify flag-transitive 2-designs. We have come to the conclusion that if <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"><a:mi mathvariant="script">D</a:mi></a:math> is a nontrivial 2-design having block size 5 and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" id="M3"><d:mi>G</d:mi></d:math> is a two-dimensional projective special linear group which acts flag-transitively on <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" id="M4"><f:mi mathvariant="script">D</f:mi></f:math> with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M5"><i:mi>q</i:mi><i:mo>≢</i:mo><i:mn>1</i:mn></i:math> (mod 4), then <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M6"><k:mi mathvariant="script">D</k:mi></k:math> is a 2-(11, 5, 2) design, a 2-(11, 5, 12) design, a 2- <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M7"><n:mfenced open="(" close=")" separators="|"><n:mrow><n:mi>q</n:mi><n:mo>+</n:mo><n:mn>1</n:mn><n:mo>,</n:mo><n:mn>5</n:mn><n:mo>,</n:mo><n:mn>2</n:mn><n:mrow><n:mfenced open="(" close=")" separators="|"><n:mrow><n:mi>q</n:mi><n:mo>−</n:mo><n:mn>1</n:mn></n:mrow></n:mfenced></n:mrow></n:mrow></n:mfenced></n:math> design with <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M8"><v:mi>q</v:mi><v:mo>≡</v:mo><v:mn>3</v:mn></v:math> (mod 4) or a 2- <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M9"><x:mfenced open="(" close=")" separators="|"><x:mrow><x:mi>q</x:mi><x:mo>+</x:mo><x:mn>1</x:mn><x:mo>,</x:mo><x:mn>5</x:mn><x:mo>,</x:mo><x:mrow><x:mfenced open="(" close=")" separators="|"><x:mrow><x:mi>q</x:mi><x:mo>−</x:mo><x:mn>1</x:mn></x:mrow></x:mfenced></x:mrow><x:mo>/</x:mo><x:mn>3</x:mn></x:mrow></x:mfenced></x:math> design with <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" id="M10"><fb:mi>q</fb:mi><fb:mo>=</fb:mo><fb:msup><fb:mrow><fb:mn>2</fb:mn></fb:mrow><fb:mrow><fb:mi>f</fb:mi></fb:mrow></fb:msup></fb:math> (where <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M11"><hb:mi>f</hb:mi><hb:mo>></hb:mo><hb:mn>2</hb:mn></hb:math> is an even).