Transformation completeness properties of SVPC transformation sets (Q1179190)

A set \(T\) of permutations of a finite set \(D\) is said to be transformation complete if the orbits of the group \(\langle T\rangle\), the group generated by \(T\), acting on the power set of \(D\) are exactly the sets of subsets of \(D\) having the same cardinality. Thus -- expressed in standard terminology -- \(T\) is transformation complete if and only if \(\langle T\rangle\) is \(k\)-fold homogeneous for all \(k\), \(0\leq k\leq| D|\). All such groups \(G\) have been determined by \textit{R. A. Beaumont} and \textit{R. P. Peterson} [Can. J. Math. 7, 35--42 (1955; Zbl 0064.02504)]; if \(| D|>9\) then \(G\) is alternating or symmetric. The authors give some generating systems \(P^n_r\) of permutations of degree \(2^n\), called `suppressed variable permutation and complementation (SVPC)', in terms of Boolean functions. They show that \[ \hbox{Sym}(2^n)=\langle P^n_{n-1}\rangle=\langle P^n_{n-2}\rangle>\langle P^n_{n-3}\rangle=\ldots=\langle P^n_1\rangle=\hbox{Alt}(2^n); \] hence \(P^n_r\) is transformation complete for \(n>r\geq 1\).