Automorphic subsets of the \(n\)-dimensional cube. (English) Zbl 0968.05039

Let \(Q_n\) be the \(n\)-dimensional cube, with vertex set \(V\) and automorphism group \(E_n\). (We think of \(V\) as the set of points in \(n\)-dimensional space whose coordinates are all \(0\) or \(1\).) The authors define an automorphic set to be a subset \(X\) of \(V\) with the property that the stabilizer of \(X\) in \(E_n\) acts transitively on \(X\). They derive some properties of automorphic sets, proving, for example, that if \(X\) is an automorphic set in \(Q_n\), then \(|X|\) divides \(2^n n! = |E_n|\) and \(|X|\leq 2^n\). They also show that these conditions are sufficient if \(n\leq 4\), but not in general. Automorphic sets are related to the “cwatsets” of G. J. Sherman and M. Wattenberg [Math. Mag. 67, No. 2, 109-177 (1994; Zbl 0835.05088)]. The authors prove that a nonempty set is a cwatset if and only if it is an automorphic set containing the vertex \(\bar{0}=(0,0,\ldots,0)\). This identification is then used to answer (in the negative) some questions about cwatsets raised by Sherman and Wattenberg.


05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20E22 Extensions, wreath products, and other compositions of groups


Zbl 0835.05088
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