372 POPULAR SCIENCE MONTHLY. 



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 ever3i;hing unchanged 

 (known as the identity) are all the possible movements of the plane 

 which transform the given triangle into itself. Hence these six move- 

 ments 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 con- 

 sist 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 ah -f- cd 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 2n movements of its plane, viz., n — 1 movements 

 around its center and n rotations through 180° around its lines of 

 symmetry, in addition to the identity. The first n — 1 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 2n 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- 



