Groups in expository literature
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Groups have received considerable attention in expository literature, aimed at a general populace with no formal mathematical training. This article looks at the different ways in which the concept of group is introduced in such literature, including fiction and non-fiction.
Popular expositions typically begin by looking at symmetries of some typical structures. For instance, the symmetries of the square, the symmetries of the cube, the operations that can be done in puzzles like the fifteen puzzle and the Rubik cube, are common starting points.
Many popular expositions on groups focus on the same issues that are considered in the survey article understanding the definition of a group. These are also the issues that mathematicians had to grapple with to progress from their intuitive understanding of groups to a formal definition.
Some of the points brought out are:
- A group needs to be closed under the multiplication.
- The axioms need to be explicitly introduced: associativity, identity element and inverses.
- A group needs to be thought of as an abstract set with a binary operation, with a particular action of the group being merely incidental. The same group can act in very different ways.