ABSTRACT

34.1 Even when the notion of an abstract group became common in many areas of mathematics, group was still used as an abbreviation for “transformation group”. The geometric connotation remained dominant until the success of Galois Theory, where groups appear as groups of automorphisms of algebraic structures, really lead to abstract groups. Then we may observe the actions of groups in “nature” and view these actions as representations of abstract groups. For example, groups may act as groups of permutations e.g. when one tries to work out all possible ways five people can take places in a Japanese car. Or one can think of groups as groups of symmetry of some solid bodies in space e.g. symmetries of regular polyhedra.