ABSTRACT
Section IV in a Nutshell This section defines the abstract notion of group, after examining some important examples: symmetries of regular n-sided polygons (called the nth dihedral groups, Dn), symmetries of the regular tetrahedron in 3-dimensional space, and most importantly, the permutations of a set of n objects, which we call the symmetric group on n, Sn. (We mostly confine ourselves to permutations of a finite set, but the collection of permutations of infinite sets are also groups.) The dihedral groups and the symmetries in space can all be thought of as subgroups of the permutations of the vertices of the corresponding geometric objects. The dihedral group Dn has 2n elements, and the symmetric group Sn has n! elements.
