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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 303-323 (2000)
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Automorphic Subsets of the $n$-dimensional Cube

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Gareth Jones, Mikhail Klin, Felix Lazebnik

Department of Mathematics, University of Southampton, Southampton SO17 1BJ, UK, e-mail: gaj@maths.soton.ac.uk; Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, P. O. Box 653, 84105 Beer-Sheva, Israel, e-mail: klin@cs.bgu.ac.il; Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA, e-mail: fellaz@math.udel.edu

**Abstract:** An automorphic subset of the $n$-dimensional cube $Q_n$ is an orbit of a subgroup of $ Aut(Q_n)$, acting on the vertices. We develop a theory of such subsets, and we show that those containing $0$ coincide with the cwatsets introduced by Sherman and Wattenberg in response to a statistical result of Hartigan. Using this characterisation, together with results from finite group theory and number theory, we answer two questions on cwatsets posed by Sherman and Wattenberg, and we complete the proofs of some results outlined by Kerr.

**Keywords:** automorphic subset, cwatset, Hamming distance, $n$-dimensional cube, permutation group, subgraph, wreath product

**Classification (MSC2000):** 05C25, 20B25, 20E22

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