The reverse representation problem

Abstract Cayley’s theorem tells us that all groups $$\textbf{G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> occur as subgroups of the group of permutations over some set X . In this paper we consider a ‘sort-of’ converse to this question: given a set X and some transformation group $$\textbf{S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> over X , what are the possible group structures on X that result in groups represented by $$\textbf{S}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> ? We solve this problem in the more general setting of faithful semigroups and observe that the solutions to this problem, which we term unrepresentations , have an inherent group structure. We study this phenomenon in depth before finishing with an analysis of the special case of unrepresentations of Clifford semigroups.