Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

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GROUPS OF PLANE FIGURES

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metric group of degree n. It is not necessary that each of
these substitutions should actually involve all of these n letters.
For instance, it may be found by trial that the two substitutions abc and abd generate a group of order 12, while the
two substitutions abc and ad generate the symmetric group of
order 24. There is no upper limit for the order of a group which
can be generated by two substitutions if these substitutions
be chosen arbitrarily and their degrees are not limited.
A set of X substitutions si, S2,..., sx of a finite substitution group G is called a set of generators of G provided there is
no subgroup in G which includes each of these substitutions.
When these substitutions satisfy the additional condition that
G can be generated by no X-1 of them, the set is said to be a
set of independent generators of G. Such a set can usually
be chosen in many different ways.
4. The Groups of Movements of Plane Figures. The symmetric group of order 6 and the octic group are special cases of
the groups of movements of regular polygons. The regular
polygons of n sides are evidently transformed into themselves
by the cyclic group of order n which is generated by the substitution corresponding to the permutation of the vertices
when the polygon is rotated around the center through the
angle 27r/n. They are also transformed into themselves by n
substitutions of order 2 which correspond to the permutation
of the vertices when the polygons are rotated successively
through the angle 7r around their different lines of symmetry.
As no other movements transform these polygons into themselves, it results that the group of movements of a regular
polygon of n sides is of order 2n.
According to a common definition of regular polygons there
is only one regular polygon of 3, 4, or 6 sides, but there
are two regular polygons of five sides, as may be seen by
connecting alternate vertices, and there are three such polygons of 7 sides. In fact, it is not difficult to see that the number of such regular polygons of n sides is equal to one-half the
number of generating substitutions of the cyclic group of order
n. All of these regular polygons of n sides belong to the same