Congruences Derived from Group Action

Since the beginning of this century the development of group theory has been dominated by the notion of representation, and the seemingly more specialized theory of group actions (permutations) has been given short shrift. To be sure, every action of a group can be considered as a particular representation by matrices, but in this setting some of the finer structure of the original permutations is lost. There are signs that the tables may be turning. Topology, ergodic theory, combinatorics and sundry other subjects abound with problems that cannot be dismissed by group characters alone. To name but one instance, the Burnside algebra yields more structural information than the Grothendik ring, as Solomon was first to note [11]. An important problem in combinatorics is to enumerate unlabeled objects, i.e. equivalence classes of "labeled" objects under a group of automorphisms. Since Polya 's fundamental paper of 1937 [6], it has been wrongly believed that all the necessary information is given by the cycle index of a permutation group. However, in the same year, Witt's enumeration of the dimensions of free Lie algebras [13] displayed the need for more detailed invariants. The same need arose in Rota's generalization of Spitzer's probabalistic formula [8]. In both these instances the problem is that of enumerating aperiodic elements of a group action (definition below) and this lies beyond the scope of the cycle index. The idea for the solution of this problem bears a resemblance to Galois theory, an approach first fully presented by Rota and Smith [9]. Whenever a group G acts on a set , one can define certain special subgroups associated with this action which we call periodic subgroups (vide below). The periodic subgroups form a lattice, in general smaller than the lattice of all subgroups of G, and this lattice is analogous to the lattice of normal subfields of a field extension. Mobius inversion over the lattice of periods gives an explicit expression for the number of aperiodic functions on the underlying set . One obtains, as a special case, proofs of congruences due to Fermat, Lucas and others. In this paper we show that similar congruences can be derived for any group of permutations whatsoever (Theorem 3.2 below). We are also led to define an analog of the Euler </J function for a general group action and to derive corresponding congruences. We surmise that other number theoretic functions can also be so generalized. Thus it is seen that the lattice of periods may be a useful enumerative invariant of a group action.