Block-transitive 3-( v , k , 1) designs associated with alternating groups

Let 𝒟 be a nontrivial 3-( v , k , 1) design admitting a block-transitive group G of automorphisms. A recent work of Gan and the second author asserts that G is either affine or almost simple. In this paper, it is proved that if G is almost simple with socle an alternating group, then 𝒟 is the unique 3-(10, 4, 1) design, and G=PGL(2,9) , M_10 or Aut(A_6 )=S_6:Z_2 , and G is flag-transitive.