Almost simple groups with socle acting on Steiner quadruple systems

Let N = L n ( q ) , n ⩾ 2 , q a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group G with N ⩽ G ⩽ Aut ( N ) . In particular, we show that G cannot act as a group of automorphisms on any Steiner quadruple system for n > 2 .