On the Spectrum of n Quasigroups with Given Conjugate Invariant Subgroup

We show in this paper that for any group G of permutations of an (n + 1) set, any composite q > (n + 1)’ is the order of an n quasigroup Q whose conjugate invariant subgroup is precisely G. If y1 is a positive integer, and X is a finite set of cardinality IX/ = q, we say that Q is an n quasigroup on X of order q if Q is a (4”) X (n $ 1) array, with entries from X, with the property that each of the y1 f 1 qn x n subarrays of Q consists of q” distinct rows. Thus each of the q” possible n tuples with entries from X occurs exactly once as a row in such a subarray. We consider two n quasigroups on X equivalent if a permutation of the rows of one produces the other. If Q is an n quasigroup on X, we denote by r(Q) the set of permutations of the columns of Q which produce an array equivalent to Q. Evidently, r(Q) is a subgroup of the symmetric group S of permutations of (0, l,..., n}. We call T(Q) the conjugate invariant subgroup of Q. These have been studied previously (see, for example, [6, 71). For each subgroup G of S, we define A(G), the spectrum of G1 to be the set of all positive integers q for which there exists an it quasigroup Q of order q with r(Q) = G. We are concerned here with the following: