Groups

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tags: - colorclass/functional analysis ---see also: - Group Theory - Group Actions - Semigroup

A group is a fundamental concept in Abstract Algebra, serving as the cornerstone for much of mathematical structure and theory. It generalizes and extends the idea of algebraic operations beyond the familiar realms of numbers to a wide array of mathematical objects. Understanding groups provides insight into symmetry, transformations, and the inherent structure of mathematical entities.

Definition

A group $(G,\cdot )$ consists of a set $G$ together with a binary operation $\cdot :G\times G\to G$ that satisfies the following four axioms:

1. Closure: For all $a,b\in G$, the result of the operation $a\cdot b$ is also in $G$. 2. Associativity: For all $a,b,c\in G$, the equation $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds. 3. Identity Element: There exists an element $e\in G$ such that for every $a\in G$, the equation $e\cdot a=a\cdot e=a$ holds. 4. Inverse Elements: For each element $a\in G$, there exists an element $b\in G$ (often denoted as $a^{-1}$) such that $a\cdot b=b\cdot a=e$, where $e$ is the identity element of $G$.

Types of Groups

- Abelian (Commutative) Groups: A group $(G,\cdot )$ is abelian if $a\cdot b=b\cdot a$ for all $a,b\in G$. Example: The set of integers $\mathbb{Z}$ with addition. - Non-Abelian Groups: A group is non-abelian if there exists at least a pair of elements $a,b\in G$ such that $a\cdot b\ne b\cdot a$. Example: The set of all invertible $2\times 2$ matrices with matrix multiplication. - Finite and Infinite Groups: A group is finite if $G$ has a finite number of elements. Otherwise, it is infinite. Example of a finite group: The set of symmetries of a square. Example of an infinite group: The set of all integers $\mathbb{Z}$ with addition.

Importance and Applications

- Symmetry: Groups are used to study symmetry in mathematics and physics. The concept of a group underlies the structure of crystals, the possible movements in space, and the symmetries of equations. - Algebraic Structures: Groups provide the framework for more complex algebraic structures like rings, fields, and vector spaces, each adding more operations or properties to the foundational group structure. - Number Theory: Groups arise naturally in the study of modular arithmetic, the solutions to polynomial equations, and the Galois theory, which connects field theory and group theory. - Topology: Groups are used to classify topological spaces through the study of continuous transformations. - Quantum Mechanics: The theory of groups underpins the symmetries and laws governing the microscopic world, including the classification of elementary particles.

Fundamental Concepts

- Subgroup: A subset $H$ of a group $G$ that is itself a group under the same operation is called a subgroup of $G$. - Cyclic Groups: A group generated by a single element, where every element of the group can be written as the generator raised to some power (or its inverse in the case of groups under multiplication). - Group Homomorphisms: Functions between groups that preserve the group operation, leading to concepts of isomorphism (structure-preserving bijection) and automorphism (isomorphism from a group to itself).

Groups are not just abstract mathematical constructs but are deeply connected to the symmetries and structures observed in nature, underpinning the laws of physics, the properties of chemical compounds, and the shape and form of geometrical objects.