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It is known that there is a finite affine (or projective) plane of order n if and only if there is a sharply two-transitive set of permutations of degree n. This paper deals with a generalization of this theorem, in which finite sets are replaced by isolated sets, cardinalities by isols and certain effectiveness conditions are imposed on the two systems involved which are trivially satisfied in the finite case. Subject classification (Amer. Math. Soc. (MOS) 1970): primary 02F40, secondarydoi:10.1017/s1446788700021108 fatcat:3rnz4d2ijzcjfgp64qryx6xuzm