The quarter turns generate a group of this order which acts on the set of components of the cube. To solve this puzzle, ie to restore a jumbled cube to its home position, you need to know the complicated laws which the generators satisfy. Indeed the structure on which it acts is the only thing we can use to give an explicit description . The semantics gives a kind of tally of the syntax, and then this group may be characterised quite straightforwardly in the language of group theory. It was originally thought that the following result, Cayley's Theorem , would eliminate the need for the abstract study of groups.
Every group has a faithful action of each variance.
Both the abstract and concrete approaches are needed to complement one another. Likewise, the beautifully simple but powerful theory of matrix representations of finite groups relies precisely upon considering all such representations. Again this supports the thesis that what mathematical objects do is more important than what they are . The Rubik cube group was specified by means of its six generators and their action on the cube, from which we could see the laws or relations . Sketches Let us transfer these ideas from groups to categories, starting with generators and laws.