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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.

MSC:

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

Citations:

Zbl 0835.05088
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