On reverse representation
Arthur Cayley’s representation theorem shows how every group \(G\) can be realized as a subgroup of the group of automorphisms over its underlying set. In this presentation we will explore some topics which focus on the inverse to this concept. If we start by considering a group \(G\) of transformations of some set \(X\), what are those groups on \(X\) which will give us \(G\) after application of Cayley’s theorem? We will investigate the unrepresentations of \(G\). It turns out that one is able to define, and characterize, the unrepresentations of various transformation structures. We will place special emphasis on transformation monoids and (certain) transformation semigroups. These unrepresentations are induced by bijections with certain properties. The properties in question end up being equivalent to our bijections being isomorphisms of two specific semigroup actions which come from our original transformation structure. An important consequence that this classification allows is that we will now be able to show that the collection of unrepresentations of any fixed structure forms an algebraic structure known as a heap. This heap is closely linked to the group of internal symmetries of any unrepresentation, a relationship which will allow us to progress by extending the concept of invertible elements to semigroups and link these back to unrepresentations. Finally, we will consider some interesting examples and non-examples of representations and unrepresentations. Doing this will allow us to make use of our theoretical work and prompt us to formulate some results with regards to finding the unrepresentations of certain structures.