Symmetry groups of Boolean Functions

TL;DR

This paper proves that most abelian permutation groups, including a broader class than previously known, can be realized as the symmetry groups of Boolean functions, solving a longstanding problem.

Contribution

It extends the class of groups known to be symmetry groups of Boolean functions, covering all groups contained in direct sums of regular groups.

Findings

All abelian permutation groups except known exceptions are symmetry groups of Boolean functions.

The result applies to a larger class of groups, including those in direct sums of regular groups.

This solves an open problem posed by Clote and Kranakis.

Abstract

We prove that every abelian permutation group, but known exceptions, is the symmetry group of a boolean function. This solves the problem posed in the book by Clote and Kranakis. In fact, our result is proved for a larger class of groups, namely, for all groups contained in direct sums of regular groups.