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The limitations of group multiplication tables: Group structure through extension theory

In Mathematical note 3356 (Gazette No. 405, October 1974) George Berzsenyi produces new groups from old by redefining multiplication in an appropriate fashion. Though the group axioms are easily checked relative to the new multiplication, the structures of the new groups are not immediately clear. This is a common situation when groups are presented by their multiplication tables which, in the present author’s view, are of little use. They do not give a clear picture of how the groups ‘work’, as anyone who has tried to verify the associative law from a table knows to his cost. Indeed, a multiplication table of a group is usually constructed only after its structure has been worked out by some other means. Take for example the non-commutative group of order 6. Much more informative than a multiplication table are its abstract presentation in terms of generators and defining relations, and its concrete realisations as the group of all permutations on three symbols and the group of isometries of the plane that map an equilateral triangle onto itself.