Definitions and Examples in Groups in Abstract Algebra

Introduction to the group

In the world of mathematics, especially in the field of abstract algebra, the concept of a group plays an important role. A group is a set of elements combined with an operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. Through the lens of group theory, abstract structures can be analyzed and understood in practical ways.

Basic definition of a group

A group is a pair (G, *), where G is a set, and * is a binary operation on G (that is, * takes any two elements a and b in G and returns an element a * b in G) that satisfies these four properties:

1. Closure: For any a, b in G, a * b is also in G
2. Associativity: For any a, b, c in G, (a * b) * c = a * (b * c)
3. Identity element: There exists an element e in G such that for every a in G, e * a = a * e = a.
4. Inverse element: For every a in G, there is an element b in G such that a * b = b * a = e, where e is the identity element.

Examples of groups

Example 1: Integers under addition

Consider the set of integers Z with the addition operation (Z, +). We have to check all the properties to confirm that (Z, +) is a group.

• Closed: Given any integers a and b, the sum a + b is also an integer. Thus, the set is closed under addition.
• Associativity: For any integers a, b, and c, we have (a + b) + c = a + (b + c).
• Identity element: The integer 0 serves as the identity element because for any integer a, a + 0 = 0 + a = a.
• Inverse elements: For any integer a, -a is an integer and a + (-a) = (-a) + a = 0.

Example 2: Real numbers under multiplication (without zero)

Consider the group of nonzero real numbers R* under multiplication, denoted (R*, .). We investigate group properties here.

• Closure: For any two nonzero real numbers a and b, the product a * b is a nonzero real number.
• Associativity: For any real numbers a, b, c, (a * b) * c = a * (b * c)
• Identity element: The number 1 serves as the identity because a * 1 = 1 * a = a for any real number a.
• Inverse element: For any non-zero real number a, there exists an inverse 1/a such that a * (1/a) = (1/a) * a = 1.

Example 3: Symmetric group

The symmetric group S_n consists of all permutations of a finite set of n elements. Let us consider the symmetric group, S_3 on three elements.

For the set {1, 2, 3}, an element of S_3 is a permutation such as (1 2 3) -> (3 1 2). The operations in S_3 are function compositions.

• Closure: The combination of any two permutations produces another permutation.
• Associativity: The task structure is associative.
• Identity element: The identity permutation leaves the elements unchanged, (abc) -> (abc).
• Inverse elements: Every permutation has an inverse that restores the order.

Visual representation

To better understand the structure of groups, imagine the complex and interesting world of symmetries in geometric objects. Let's imagine a basic example involving group symmetries:

Example 4: Symmetries of an equilateral triangle

Consider an equilateral triangle with vertices A, B and C. Its symmetries, such as rotation and reflection, form a group called the dihedral group D_3.

- Identity (e): No change. 
- Rotation by 120° clockwise (r1) 
- Rotation by 240° clockwise (r2) 
- Reflection over axis through vertex A (s1) 
- Reflection over axis through vertex B (s2) 
- Reflection over axis through vertex C (s3)

We can see the rotation and the image.

With visualization, group operations become easier to understand and can help make connections to more abstract algebraic concepts.

Advanced concepts in group theory

Now, let's look at some more advanced concepts arising from the basic definition of groups.

Subgroups

A subgroup is a subgroup of a group that is itself a group under the same operation. If H is a subgroup of G, we often write H ≤ G

For example, consider the subgroup of even integers 2Z under addition, which is a subgroup of the integers Z. Check that 2Z obeys the group rules under addition: closure, associativity, identity (zero), and inverse (negative).

Cyclic groups

A group is called a cyclic group if there exists an element g in the group such that every element of the group can be expressed as some power of g (using the group operation). The element g is called the generator of the cyclic group.

For example, consider the integers modulo n, denoted by Z_n. A cyclic group is formed for any integer a with gcd(a, n) = 1. In Z_6, the elements 1 and 5 can be chosen to form the group.

Homeomorphisms

A homomorphism is a structure-preserving map between two groups. If (G, *) and (H, +) are groups, then the homomorphism f: G → H satisfies the following property:

f(a * b) = f(a) + f(b)

This property means that the operations performed in the first group are respected by the map of the second group.
Consider the group of integers under addition (Z, +) and the homomorphism f: Z → Z defined by f(n) = 2n. Check:

f(a + b) = 2(a + b) = 2a + 2b = f(a) + f(b)

Quotient group

A quotient group is built from a group G and a normal subgroup N of it. The quotient group, written as G/N, consists of the cosets of N in G

Let Z be the group of integers under addition, and 2Z be the subgroup of even integers. The quotient group Z/2Z has two cosets: the even coset, and the odd coset. Here, you can think of Z/2Z as equivalent to Z_2, which is the integers modulo 2.

Conclusion

The study of groups in abstract algebra reveals the underlying symmetric and structural components that form the backbone of many mathematical concepts. Whether addressing permutations of objects, symmetries in geometry, or the structure of mathematical entities in arithmetic, the beauty of group theory lies in its ability to unify and generalize disparate mathematical situations into a coherent field of study.

Through the examples given, including integers, real numbers, and geometric symmetries, you can see the diverse applications of group theory in representing simple and complex mathematical phenomena. This diversity underlines the fundamental role of group theory in modern algebra and highlights the importance of its properties in understanding mathematical structures in depth.

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