## 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

### Keywords:

automorphic set; cwatset; $$n$$-cube

Zbl 0835.05088
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