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.