91Ó°ÊÓ

Complex numbers are numbers that can be expressed in the form \(a + ib\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). They form a two-dimensional number space. This unique property allows us to represent complex numbers on a plane, often referred to as the complex plane or Argand diagram. Here, the horizontal axis represents the real part, \(a\), and the vertical axis represents the imaginary part, \(b\).

Complex numbers extend the concept of one-dimensional numbers into two dimensions but with addition, subtraction, multiplication, and division following special rules based on the properties of \(i\). A complex number's magnitude or modulus is given by \( \sqrt{a^2 + b^2} \). The complex conjugate of \(a + ib\) is \(a - ib\). Conjugation essentially "flips" the complex number across the real axis, which is a crucial operation when dealing with complex numbers, and specifically, automorphisms in complex number spaces.

Group theory is an area of mathematics that studies algebraic structures known as groups. A group \(G\) is a set accompanied by an operation \(*\) that combines any two elements to form a third element, and this operation satisfies four primary conditions:

• Closure: For any \(a, b \in G\), the result of the operation, \(a * b\), is also in \(G\).
• Associativity: For any \(a, b, c \in G\), \((a * b) * c = a * (b * c)\).
• Identity Element: There is an element \(e \in G\) such that for every \(a \in G\), the equation \(e * a = a * e = a\) holds.
• Inverse Element: For each \(a \in G\), there is an element \(b \in G\) such that \(a * b = b * a = e\), where \(e\) is the identity element.

Groups can describe many fundamental operations in mathematics, such as addition of integers, multiplication of non-zero real numbers, and, as our exercise involves, the multiplication of non-zero complex numbers \(\mathbb{C}^*\). Within \(\mathbb{C}^*\), the group operation is multiplication, and non-zero complex numbers satisfy all group conditions. Automorphisms within this group are transformations that preserve these group structures.

Injective and surjective functions are significant in the study of functions, transformations, and automorphisms. An injective function, or one-to-one function, is a function where each element of the function's domain (input) maps to a distinct and unique element of the range (output). In simple terms, different inputs always yield different outputs.

A function \(f: A \to B\) is injective if \(f(a_1) = f(a_2)\) implies \(a_1 = a_2\) for any \(a_1, a_2 \in A\). In the context of our exercise, we saw that if two complex numbers \(z_1\) and \(z_2\) map to the same image, then \(z_1 = z_2\), thus proving injectivity.

Surjective functions, on the other hand, cover every element of the target set. A function \(f: A \to B\) is surjective if for every element \(b \in B\), there is at least one \(a \in A\) such that \(f(a) = b\). Surjectivity in the context of our exercise was shown by demonstrating that every element in \(\mathbb{C}^*\) has a corresponding pre-image under the given transformation. When a function is both injective and surjective, it is called bijective, forming a one-to-one correspondence between elements of the domain and range, critical in defining automorphisms.