Block-transitive 3-designs with affine automorphism group

Abstract

A t-(v, k, λ) design is a pair D = (X,B) where X is a set of v points and B is a set of k-subsets of X called blocks such that any t points are contained in exactly λ > 0 blocks. If B contains all k-subsets of X then D is said to be trivial. Let G = AutD = {g ∈ Sym(X) | B = B}. That is, G is the set of all permutations of X that fix B setwise. The group G is said to be block-transitive if B = B for B ∈ B, where B = {B | g ∈ G}. In this case we say that D is block-transitive. Similarly, G is point-transitive if X = x for some x ∈ X, where x = {x | g ∈ G}.