There exists a simple non trivial t-design with an arbitrarily large automorphism group for every t
In 1987, Teirlinck proved that if t and upsilon are two integers such that upsilon = t(mod(t + 1)\((2t+1))) and upsilon greater than or equal to t + 1 > 0, then there exists a t - (upsilon, t + 1, (t + 1)\((2t+1))) design. We prove that if there exists a (t + 1) - (upsilon, k, lambda) design and a t - (upsilon - 1, k - 2, lambda (k - t - 1)/(upsilon - k + 1)) design with t greater than or equal to 2, then there exists a t - (upsilon + 1, k, lambda(upsilon - t + 1)(upsilon - t)/ (upsilon - k + 1)(k - t)) design. Using this recursive construction, we prove that for any pair (t, n) of integers (t greater than or equal to 2 and n greater than or equal to 0), there exists a simple non trivial t - (upsilon, k, lambda) design having an automorphism group isomorphic to Z(2)(n).
t-design automorphism group and recursive construction