Flag-transitive 2-designs with prime-square or prime-cube block length

We study 2- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (v,k,λ) designs <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">D</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">P</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> D=(P,B) admitting a flag-transitive automorphism group 𝐺, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> k=p² or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> p³ , and 𝑝 is prime. We prove that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> k=p² , then 𝐺 must be point-primitive, and 𝐺 is of affine or almost simple type except for two examples. If <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> k=p³ , and 𝐺 is point-primitive, then 𝐺 is of affine, almost simple, or product type with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mi>H</m:mi> <m:mo lspace="0.222em" rspace="0.222em">≀</m:mo> <m:msub> <m:mi>S</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> G≤ H S₂ acting on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">P</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:mrow> </m:math> P=Δ×Δ , where 𝐻 is a primitive group on Δ. Furthermore, if 𝐺 is point-imprimitive with a system Σ of imprimitivity consisting of 𝑑 classes of size 𝑐, we determine the parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (v,k,μ,c,d) , where 𝜇 is the size of the nonempty set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>∩</m:mo> <m:mi>C</m:mi> </m:mrow> </m:math> B∩ C with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>B</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">B</m:mi> </m:mrow> </m:math> B and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Σ</m:mi> </m:mrow> </m:math> C∈Σ . Moreover, some infinite families of flag-transitive 2- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi>λ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (v,p³,λ) designs are also constructed.