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Equivalence classes of functions between finite groups. (English) Zbl 1376.20030
Summary: Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function $$f$$, calculated from the image of its coboundary $$\partial f$$. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if $$f$$ is a homomorphism, and takes its maximum value if the graph of $$f$$ contains a splitting RDS.

##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D99 Abstract finite groups 05E15 Combinatorial aspects of groups and algebras (MSC2010) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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