Groups of Isometries In this project we investigate the relationship between complex analy...

Groups of Isometries In this project we investigate the relationship between complex analysis and the Euclidean geometry of the Cartesian plane.

The Euclidean distance between two points (x1, y1) and (x2, y2) in the Cartiesian plane is

Of course, if we consider the complex representations z1 = x1 +iy1 and z2 = x2 +iy2 of these points, then the Euclidean distance is given by the modulus

A function from the plane to the plane that preserves the Euclidean distance between every pair of points is called a Euclidean isometry of the plane. In particular, a complex mapping w = f(z) is a Euclidean isometry of the plane if

for every pair of complex numbers z1 and z2.

(a) Prove that every linear mapping of the form f(z) = az + b where |a| = 1 is a Euclidean isometry.

A group is an algebraic structure that occurs in many areas of mathematics. A group is a set G together with a special type of function ∗ from G × G to G. The function ∗ is called a binary operation on G, and it is customary to use the notation a ∗ b instead of ∗(a, b) to represent a value of ∗. We now give the formal definition of a group. A group is a set G together with a binary operation ∗ on G, which satisfies the following three properties:

(i) for all elements a, b, and c in G, a ∗ (b ∗ c) = (a ∗ b) ∗ c,

(ii) there exists an element e in G such that e ∗ a = a ∗ e = a for all a in G, and

(iii) for every element a in G there exists an element b in G such that a ∗ b = b ∗ a = e. (The element b is called the inverse of a in G and is denoted by a−1.)

Let Isom+(E) denote the set of all complex functions of the form f(z) = az + b where |a| = 1. In the remaining part of this project you are asked to demonstrate that Isom+(E) is a group with composition of functions as the binary operation. This group is called the group of orientation-preserving isometries of the Euclidean plane.

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(b) Prove that composition of functions is a binary operation on Isom+ (E). That is, prove that if f and g are functions in Isom+ (E), then the function f ◦ g defined by f ◦ g(z) = f(g(z)) is an element in Isom+ (E).

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(c) Prove that the set Isom+ (E) with composition satisfies property (i) of a group.

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(d) Prove that the set Isom+ (E) with composition satisfies property (ii) of a group. That is, show that there exists a function e in Isom+ (E) such that e ◦ f = f ◦ e = f for all functions f in Isom+ (E).

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(e) Prove that the set Isom+ (E) with composition satisfies property (iii) of a group.