On the relational complexity of a finite permutation group

Abstract

The relational complexity $$\rho (X,G)$$?(X,G) of a finite permutation group is the least k for which the group can be viewed as an automorphism group acting naturally on a homogeneous relational system whose relations are k-ary (an explicit permutation group theoretic version of this definition is also given). In the context of primitive permutation groups, the natural questions are (a) rough estimates, or (preferably) precise values for $$\rho $$? in natural cases; and (b) a rough determination of the primitive permutation groups with $$\rho $$? either very small (bounded) or very large (much larger than the logarithm of the degree). The rough version of (a) is relevant to (b). Our main result is an explicit characterization of the binary ($$\rho =2$$?=2) primitive affine permutation groups. We also compute the precise relational complexity of $${{\mathrm{Alt}}}_n$$Altn acting on k-sets, correcting (Cherlin in Sporadic homogeneous structures. In: The Gelfand Mathematical Seminars, 1996---1999, pp. 15---48, Birkhauser 2000, Example 5).