Ed's post reminded me of one huge area of ignorance of mine (among many): Given a set S having n^2 elements, we can of course arrange them in an n x n square and pick out two collections R, C, of subsets of size n: the rows and the columns. These have the nice property that for any permutations P_r and P_c of R and C, respectively, AND for any (i.e., either) permutation P_rc of {R,C}, there is a permutation P of S that simultaneously implements P_r, P_c and P_rc (meaning the composition of these). ***QUESTION: For a given n, are the collections R and C *maximal* with respect to this kind of symmetry? Or a bit more flexibly, is 2 the largest number of collections of subsets of S of size n that can have this kind of hierarchical symmetry? (E.g., can there be 3 collections of subsets of S: (X_1, X_2, X_3} such that for any permutations P_1, P_2, P_3 of X_1, X_2, X_3, respectively -- AND any permutation of {X_1, X_2, X_3} (among 6) -- then there is a permutation P of S that simultaneously implements these ?) (Similar questions can be asked of a set of size n^k -- arrange its elements as an n^k cube and start with the symmetry of all k subsets of parallel row-analogues (with n^(k-1) in each set) -- . . . and even go further with k^(n-r) subsets of orthogonally oriented parallel r-cubes, of size k^r -- but I'll restrain myself.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele