Permutation and Rearrangement

Résumé

This paper follows a presentation of J.M. Vappereau (Essaim, 1985) of a 3-point geometry constructed from the group of bijections of a set  of 3 elements. Following Hunter (1966), we will call these bijections “permutations” or “rearrangements”, depending on whether they stand for the transformation of a figure or for the transformation of its embedding space. In this paper, we construct a group from the set of all bijective mappings of three elements, and define rearrangement as an isomorphism on this group. In section 3, we develop our own 3-point geometry, defined as a group action on a coset space, with the above distinctions in mind. Finally, we briefly explore the implications of our construction for geometric representation generally.