From Complex Number to Groups

The geometric interpretation of comple number:

• the complex plane.

Why complex numbers is needed?

• Mathematically: oringin — the quadratic equation — no geometric meaning

  Fact — originated from cubic equation, e.g.: \(x^3 = 3px + 2q\) A solution is (Cardano's formula) \[\sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2 - p^3}}.\] if \({q^2 - p^3} < 0\), clearly complex number would apppear in the cubic roots. (e.g., p = 5, q = 2, obviously, 4 is one of the root.)

  Mathematically, \(\mathbb{R}\) is not algebraically complete but \(\mathbb{C}\) are.

• Physically: complex number render many physical expressions much simpler, is necessary to quantum physics.

For \(z = x + y \mathrm{i}\), where \(x = \Re z, y = \Im z\). We define

Complex conjugate

\[\bar{z} = x - y\mathrm{i}.\]

Modulus

\[|z| = |\bar{z}| = \sqrt{x^2 + y^2}.\]

Argumnet angle

\[ \text{Arg}\;z = \text{arg}\;z\] where \(\text{arg}\;z\) is Principle value of \(\text{Arg}\;z\), \(\text{arg}\;z\in[0,2\pi)\).

Conside compelx numbers with unit-modulus, \[z = \cos\theta + \mathrm{i}\sin\theta, \;z^{\prime} = \cos\psi + \mathrm{i}\sin\psi.\] Then we have \[zz^{\prime} = cos(\theta + \psi) + \mathrm{i} \sin(\theta + \psi).\] Anyway, we can prove the Euler formula \[\mathrm{e}^{\mathrm{i}\theta} = \sin\theta + \mathrm{i}\sin\theta.\] Consequently, any complex number can be written as \[z = |z|\mathrm{e}^{\mathrm{i}\theta},\;\theta = \text{Arg}\;z.\] Clearly that all the unit-modulus comlpex # form a circle.

Since

• \(\mathrm{i}^1 = \mathrm{i}\)
• \(\mathrm{i}^2 = -1\)
• \(\mathrm{i}^3 = -\mathrm{i}\)
• \(\mathrm{i}^4 = 1\)

and further power of i cannot generate any number beyond the set \(G_i = \{1, \mathrm{i}, -1, -\mathrm{i}\}\). Then we can called such a mathematical structure group. A group is a set of elements equipped with a multiplication: \(G\times G\mapsto G\) that is associative and statisfies the following conditions:

• Closure: \(\forall g_1, g_2\in G, g_1g_2 \in G.\)
• Unit: \(\exists e\in G, \forall g \in G, eg = ge = g\). \(e\) is called the unit or identity of \(G\).
• Inverse \(\forall g\in G, \exists g^{\prime}\in G, gg^{\prime} = g^{\prime}g = e\). Define \(g^{-1} = g^{\prime}\).

Easy to prove that there`s only one unit in a group and each element has only one inverse. Thus we have

Elimination law

\[ ab = ac \Rightarrow b = c\]

And when the group \(G\) fits

• Communation: \(\forall a, b \in G, ab = ba \).

We called \(G\) as abelian group.

Obviously, \(\mathbb{Z}, \mathbb{R}, \mathbb{C}\) is a group under addition. Group like this is called discrete group.

Finite group

A discrete group with finite number of elements is a finite group.

Order

The number of its elements is called order of the group.

Two groups \(G\) and \(H\), a map \(f: G\mapsto H\)

Group Homorphism

If \(f\) prevserves the group multiplication, i.e., \[\forall g_1, g_2 \in G, f(g_1, g_2) = f(g_1, g_2).\]

Isomorphism (\(G\cong H\))

\(f\) is a homorphism from \(G\) to \(H\), also \(f\) is a bijection. i.e., \(f^{-1}\) is a homorphism from \(H\) to \(G\).

Endomorphism

f is a isomorphism between \(G\) and \(H\), and \(H = G\).