Symmetry Groups of Boolean Functions and Constructions of Permutation Groups Original Research Article

In this paper we deal with the symmetry groupS(f) of a boolean functionfonn-variables, that is, the set of all permutations onnelements which leavefinvariant. The main problem is that of concrete representation: which permutation groups onnelements can be represented asG = S(f) for somen-ary boolean functionf. Following P. Clote and E. Kranakis [[6]] we consider, more generally, groups represented in such a way byk-valued boolean functions (i.e., functions on a two-element set withkpossible values) and call such permutation groupsk-representable (k ≥ 2). The starting point of this paper is a false statement in one of the theorems of [[6]] that everyk-representable permutation group is 2-representable for allk ≥ 2. We show that there exists a 3-representable permutation group that is not 2-representable. A natural question arises whether there are other examples like this. This question turns out to be not easy and leads us to considering general constructions of permutation groups and investigating their properties. The main result of the present paper may be summarized as follows: using known groups and “standard” constructions no other example like that above can be constructed. Nevertheless, we conjecture that there arek + 1-representable permutation groups that are notk-representable for allk ≥ 2. If this is true, then this would open an interesting avenue toward investigating and classifying finite permutation groups.