Elations of Designs

An elation of a design 2 is an automorphism y of Sf fixing some block X pointwise and some point x on X blockwise. Luneburg [4] and I [2] have proved results which state that a design admitting many dations and having additional properties must be the design of points and hyperplanes of a finite desarguesian projective space. In this note, additional results of this type will be proved and applied to yield a generalization of a previous result on Jordan groups [3]. The proofs were suggested by a result of Hering on dations of finite projective planes [1, pp. 122, 190]. Much of our notation can be found in [1]. Designs will always satisfy v è k + 2, and the blocks will be distinguishable as sets of points. Isomorphic designs will be identified. The complement of the block X is fâX. If Y is an automorphism group of a design, and x f l , then Y(X) and T(x) are the largest subgroups of T fixing X pointwise and x blockwise, respectively. If TL(X) ^ T(X), then H(x, X) = T(x) H TL(X). If U(X) ^ T(X) for all X, then, for each block X and each point x, II(X)* is the set {Jy<zX*R(y, X) and II(x)* = UzçrII(x, Y), [a, j3] is the commutator orfi~afi. If g is a power of a prime p and n is an integer, g\\n means that g\n but pg\n. A permutation group is said to act regularly if only the identity fixes a point.