ABSTRACT
In Chapters 22 and 23 we used the idea that a symmetry of a geometric object like a triangle or tetrahedron must take vertices to vertices. For example, specifying where the vertices of a tetrahedron are sent completely determines the symmetry function. We used this reasoning to determine a complete list of symmetries of the tetrahedron. We called a specification of how the vertices are moved a permutation of the vertices. We will now consider the notion of permutations in an abstract setting. This leads to an important theorem from group theory, which says that all finite groups can be thought of as groups of permutations.
Consider the list 1, 2, 3, 4, · · · , n of the first n positive integers. We wish to rearrange or permute this list. To do this, we must tell ourselves which slot the integer 1 should be placed in, which slot the integer 2 should be placed in, and so forth. What this means is that a permutation of this list amounts to a function
