On the Dimension of Finite Permutation Group Actions

Jonathan D. H. Smith

Abstract: The dimension (or ``minimal base size'') of a finite permutation group action is defined to be the smallest power of the action that contains a regular orbit. Although the concept has appeared before in various contexts, the intention of the current paper is to survey it from a slightly different viewpoint, with particular emphasis on its behaviour with respect to $G$-set constructions. Elementary inequalities relate the dimension to the degree and closure properties of the action. The dimension is also expressed exactly in terms of the Möbius function of the subgroup lattice of the permutation group. For geometric permutation actions, the dimension is related to the geometric dimension of the space being acted on. The behaviour of the dimension is studied with respect to disjoint unions, Cartesian products, and wreath products of actions. Use of the wreath product construction exhibits permutation group actions with arbitrary positive integral dimension and degree-to-dimension ratio.

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