this group is said to be cyclic. Cyclic groups are the simplest possible groups and they are the only ones whose operations can be completely represented by complex numbers.
Another very simple category of groups of operations is furnished by the totality of movements which leave a regular polygon unchanged. For instance, a regular triangle is transformed into itself when its plane is rotated around its center through 120° or through 240°. Moreover, its plane may be rotated through 180° around any of its three perpendiculars without affecting the triangle as a whole. These five rotations together with the one which leaves everything unchanged (known as the identity) are all the possible movements of the plane which transform the given triangle into itself. Hence these six movements form a group, which happens to be identical with the group formed by the six possible permutations of three things.
It is not difficult to see that a plane can have just eight movements which do not affect the location of a given square in it. These consist of the three movements around the center of the square through 90°, 180° and 270° respectively; the four movements through 180° around the diagonals and the lines joining the middle points of opposite sides; and the identity. This group of eight operations has exactly the same properties as the permutation group on four letters which transforms into itself. Hence these two groups are said to be simply isomorphic. From the standpoint of abstract groups, such groups are said to be identical.
In general, a regular polygon of n sides is left unchanged as a whole by just movements of its plane, viz., movements around its center and rotations through 180° around its lines of symmetry, in addition to the identity. The first movements together with the identity clearly form a group by themselves. Such a group within a group is known as a subgroup. This category of groups of operations is known as the system of dihedral rotation groups or the system of the regular polygon groups. It is not difficult to prove that each of them is generated by some two non-commutative rotations through 180° and that no other groups have this property.
Among the non-regular polygons the rectangle with unequal sides has perhaps the most important group. There are clearly just three movements of the plane (besides the identity) which transform such a rectangle into itself, viz., the rotation through 180° around the center and the rotation around its two lines of symmetry through the same angle. These four operations form a group which presents itself in very many problems and is known by a number of different names. Among these are the following: four-group, anharmonic ratio group, axial group, quadratic group, rectangle group, etc. Since we arrive at the identity by repeating any one of its operations, it is entirely dif-
