Some Finite and Discrete Groups

We have have encountered several finite groups in the previous chapter: the cyclic group Zn, the symmetric group Sn, and its subgroup the alternating group An. The properties of these groups will be elaborated in this chapter. We will also discuss three other groups obtained by incorporating ‘reflections’ into Zn, the dihedral group Dn, the generalized quaterion group Qn, and the dicyclic group DZn. These three groups differ from one another in the way ‘reflections’ are introduced. A large class of finite groups is obtained from matrices over a finite field. In mathematics, a field is collection of objects that can mutually multiply, divide, add, and subtract. Real numbers R and complex numbers C are two familiar fields, each having an infinite number of elements. Thus matrix groups over these fields similarly have an infinite order. Integers Z do not form a field because we cannot divide them, but integers modulo a prime number pdo form finite fields which we shall denote by Fp. As a result, matrix groups over these finite fields also have a finite orders. Let me illustrate how to carry out the four arithmetic operations in the field F7: 2 × 5 = 10 ≡ 3 (mod 7), 6 + 8 = 14 ≡ 0, 8 − 11 = −3 ≡ 4, and 6/5 ≡ 4 (because 4× 5 = 20 ≡ 6 (mod 7)). There are also finite fields Fpn with p elements, but the four arithmetic operations (+,−,×, /) are much more complicated. Finite subgroups of O(3) are symmetry groups of molecules and crystals. They are known as point groups. Some of them may also be symmetry groups for the three generations of elementary fermions. (Crystal)