V02

Title: Vorlesung 2: Gruppen, Körper, Symmetrien Mathematik I
Date: April 22, 2026
Speaker: Stephan Block
Institution: AG Bionanogrenzflächen, Institut für Physikalische und Theoretische Chemie, FB BCP, Freie Universität Berlin

Understand the main number sets and their applications.
Recognize that transformation of equations is facilitated by the (mathematical) structure of groups and fields.
Realize that addition and multiplication are specific cases of abstract operations defined on sets.

Definition: A non-empty set M and an operation ○ form a group G = (M, ○) if the following criteria are met: 1. Closure: For all a, b ∈ M, there exists c ∈ M such that a ○ b = c. 2. Associativity: For all a, b, c ∈ M, (a ○ b) ○ c = a ○ (b ○ c). 3. Identity Element: There exists a unique element e ∈ M such that for all a ∈ M, a ○ e = e ○ a = a. 4. Inverse Elements: For each a ∈ M, there exists an element a⁻¹ ∈ M such that a ○ a⁻¹ = a⁻¹ ○ a = e.
Remarks: 1. Group criteria ensure that the operation ○ always yields an element in the group. 2. If in addition the commutative law holds (a ○ b = b ○ a for all a, b ∈ M), then G is called abelian.

Definition: The set of natural numbers is defined as: ℕ = {0, 1, 2, 3,…}.
Remarks: 1. The operations of addition (+) and multiplication (∙) are defined on ℕ. These operations are associative and commutative and satisfy some, but not all, group criteria. 2. Addition and multiplication are distributive, i.e., for all a, b, c ∈ ℕ, a ∙ (b + c) = a ∙ b + a ∙ c. 3. For a, b ∈ ℕ, there exist uniquely determined numbers s, r ∈ ℕ with r < b such that a = s·b + r. The number r is called the remainder when dividing a by b, often expressed as r = a mod b.

A number is called a prime if it is divisible only by itself and 1.
Prime numbers are important because every natural number can be uniquely expressed as a product of prime numbers.
Application: The statement, "Let p be a natural number. If p² is an even number, then p is also even,” can be verified by decomposing p into its associated product of prime numbers pk. If p² is divisible by pk, then p must also be divisible by pk.
An alternative proof can begin with showing that for any odd p, p² is also odd, then apply the contrapositive.

Definition: The set of integers is defined as: ℤ = {…, -3, -2, -1, 0, 1, 2, 3,…}.
Remarks: 1. Addition and multiplication of ℕ extend naturally to ℤ. 2. The set of integers ℤ forms an abelian group (G = (ℤ, +)) with the identity element being 0. 3. For every z ∈ ℤ, there exists an additive inverse -z ∈ ℤ such that z + (-z) = 0. Moreover, (-1)·a = -a and (-1)·(-1) = 1.

Definition: The set of rational numbers is defined as: ℚ = {r/q : r, q ∈ ℤ, q ≠ 0}.
Remarks: 1. (ℚ, +) and (ℚ, ∙) are abelian groups. For every q ∈ ℚ, there exists an additive inverse -q ∈ ℚ and a multiplicative inverse q⁻¹ ∈ ℚ such that q ∙ q⁻¹ = 1. 2. Addition and multiplication from ℤ extend naturally to ℚ.

Definition: The set of real numbers is defined, incorporating both rational and irrational numbers.
Remarks: 1. The set ℚ has 'gaps'; for example, the equation x² = 2 has no solution in ℚ, illustrating the concept of irrational numbers. Other examples include π and Euler's number e. 2. The set of real numbers can be understood as the union of rational and irrational numbers. 3. Addition and multiplication from ℚ transfer naturally to ℝ.

Definition of Intervals: Important subsets of real numbers include the following intervals: - Open interval: x ∈ (a, b) - Left-open interval: x ∈ [a, b) - Right-open interval: x ∈ (a, b] - Closed interval: x ∈ [a, b]

Definition: The set of complex numbers is defined as: ℂ = {z = x + iy : x, y ∈ ℝ; i² = -1}.
Remark: The set ℂ is introduced for completeness and will be discussed in detail in the next lecture.

Conclusion: Let G = (M, ○) be a group. Since G meets the group criteria, in any equation of the form a○x = b, x can be uniquely solved for all elements a and b in M. Thus, the following operations are allowed on the previously discussed number sets: - ℕ: +, ∙: √, lim representation: partially defined - ℤ: +, ∙: √, lim representation: partially defined - ℚ: +, ∙: √, lim representation: partly defined - ℝ: +, ∙: √, lim representation: partly defined - ℂ: +, ∙: √, lim representation: fully defined - General operation: a ○ x = b ⇔ x = a⁻¹ ○ b.

The discussed number sets (ℤ to ℂ) are examples of groups with infinitely many elements. However, there are many important groups with only finitely many elements (finite groups).
A prominent example in chemistry and biochemistry are symmetry groups.

For instance, when CO₂ is rotated 180° in the 'plane', all atoms return to the same position as initially. Thus, the molecule CO₂ possesses symmetry for rotations of 180°.
Operations that map a molecule onto itself are called symmetry operations. They form a group with the operation defined by the successive application of these symmetry operations.

Besides 180° rotational symmetry (denoted d180), CO₂ possesses other symmetries such as: - Inversion center (i) - Rotational axis - Reflection planes (s)

Using the three symmetry operations (d180, i, and s), we can form a group. Executing the 180° rotation twice results in a 360° rotation, bringing every molecule back to its original position. This 'null operation', which produces no change, is denoted by 0. It represents the identity element of the group. Since performing the 180° rotation (and reflection and inversion) twice corresponds to 0, these operations (d180, i, and s) are self-inverse.

You may wonder about the practical advantage of discussing symmetries. A system possessing symmetries allows for a simplified description of that system. Furthermore, the presence of symmetries can influence measurable quantities such as vibrational bands, observable (or not) via infrared spectroscopy.