3.6 Groups

Groups are the building blocks of abstract algebra, providing a framework for studying symmetry and transformations. They're defined by four key properties: closure, associativity, identity, and inverse elements. These properties allow mathematicians to analyze patterns and relationships across various mathematical systems.

Examples of groups include integer addition and symmetries of geometric shapes. Non-examples, like natural numbers under addition, help clarify group requirements. Understanding group properties, types, and operations is crucial for grasping more complex algebraic structures and their applications in mathematics and related fields.

Definition of groups

• Groups form a fundamental algebraic structure in mathematics, providing a framework for studying symmetry and transformations
• Understanding groups enhances mathematical thinking by revealing underlying patterns and relationships in various mathematical systems
• Group theory serves as a powerful tool for problem-solving and abstraction in mathematics and related fields

Group axioms

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• Consist of four fundamental properties that define a group structure
• Closure property ensures the result of the group operation remains within the group
• Associativity property allows for flexible grouping of elements in operations
• Identity element acts as a neutral element in the group operation
• Inverse elements guarantee each element has a counterpart that "undoes" its effect

Examples of groups

• Integer addition group $(Z, +)$ includes all integers with addition as the operation
• Multiplicative group of non-zero real numbers $(R*, \times)$ consists of real numbers excluding zero with multiplication
• Symmetry groups of geometric shapes (rotations and reflections of a square)
• Matrix groups (general linear group $GL(n, R)$ of invertible n×n matrices)

Non-examples of groups

• Natural numbers under addition lack inverse elements for non-zero numbers
• Even integers under multiplication fail the closure property
• Rational numbers under division do not form a group (no identity element, not closed)
• Set of all 2×2 matrices under matrix multiplication (not all matrices have inverses)

Group properties

Closure property

• Ensures the result of combining any two group elements remains within the group
• Mathematically expressed as $\forall a, b \in G, a * b \in G$ where G is the group and * is the group operation
• Maintains the integrity and self-containment of the group structure
• Allows for consistent application of the group operation without leaving the set

Associativity property

• Guarantees the order of operations does not affect the result when combining three or more elements
• Expressed as $(a * b) * c = a * (b * c)$ for all elements a, b, c in the group
• Enables flexible grouping of elements in calculations and proofs
• Crucial for defining group structures on sets with various operations

Identity element

• Unique element e in the group that leaves other elements unchanged when combined with them
• Satisfies the equation $a * e = e * a = a$ for all elements a in the group
• Acts as a neutral element in group operations
• Examples include 0 for addition and 1 for multiplication in number groups

Inverse elements

• Each group element a has a unique inverse element b such that $a * b = b * a = e$ (e is the identity element)
• Allows "undoing" the effect of any group element
• Crucial for solving equations within the group structure
• In additive groups, inverses are often called negatives; in multiplicative groups, reciprocals

Types of groups

Finite vs infinite groups

• Finite groups contain a countable number of elements
  • Order of a finite group refers to the number of elements it contains
  • Examples include symmetric groups of finite sets, cyclic groups of finite order
• Infinite groups have an uncountable or infinite number of elements
  • Can be countably infinite (integers under addition) or uncountably infinite (real numbers under addition)
  • Require different techniques for analysis and proof compared to finite groups

Abelian vs non-abelian groups

• Abelian groups (commutative groups) satisfy the commutativity property: $a * b = b * a$ for all elements
  • Simplify many calculations and theorems in group theory
  • Examples include integer addition, real number multiplication
• Non-abelian groups do not have commutative operations
  • More complex structure, often arising in geometry and physics
  • Examples include matrix multiplication groups, symmetry groups of non-square objects

Cyclic groups

• Generated by a single element called the generator
• All elements can be expressed as powers (or multiples) of the generator
• Can be finite (integers modulo n under addition) or infinite (integers under addition)
• Simplest type of group, fundamental to understanding more complex group structures

Symmetric groups

• Consist of all bijective functions (permutations) on a set
• Denoted as $S_n$ for a set of n elements
• Fundamental in the study of permutations and combinatorics
• Non-abelian for n > 2, showcasing complex group behaviors

Group operations

Binary operations

• Functions that combine two elements of a set to produce another element of the same set
• Must be well-defined, meaning each pair of inputs yields a unique output
• Examples include addition, multiplication, and composition of functions
• Form the foundation of group structures and other algebraic systems

Group tables

• Visual representations of group operations, especially useful for finite groups
• Rows and columns labeled with group elements, entries show operation results
• Reveal patterns and properties of the group (commutativity, subgroups)
• Cayley tables provide a complete description of the group structure

Composition of elements

• Refers to the repeated application of the group operation
• Expressed as powers in multiplicative notation ($a^n$) or multiples in additive notation (na)
• Leads to important concepts like order of an element and cyclic subgroups
• Fundamental in understanding group structure and behavior

Subgroups

Definition of subgroups

• Subset H of a group G that forms a group under the same operation as G
• Must satisfy closure, associativity, identity, and inverse properties within the subset
• Inherits the group structure from the parent group
• Provides insight into the structure and properties of the larger group

Proper vs improper subgroups

• Proper subgroups are strict subsets of the group, excluding the entire group itself
  • Reveal internal structure and symmetries within the larger group
  • Examples include even integers as a subgroup of all integers under addition
• Improper subgroups include the entire group G and the trivial subgroup {e}
  • Always exist for any group
  • Serve as boundary cases in subgroup analysis

Cyclic subgroups

• Generated by a single element of the group
• Consist of all powers (or multiples) of the generating element
• Order of the subgroup determined by the order of the generating element
• Fundamental building blocks for understanding group structure

Order of subgroups

• Number of elements in the subgroup
• Lagrange's Theorem states that the order of a subgroup divides the order of the group
• Provides constraints on possible subgroup sizes
• Crucial in the study of group structure and classification

Group homomorphisms

Definition of homomorphisms

• Structure-preserving maps between groups
• Satisfy the property $f(ab) = f(a)f(b)$ for all elements a, b in the domain group
• Preserve group operations and structure
• Enable comparison and analysis of different group structures

Kernel and image

• Kernel of a homomorphism f: G → H is the set of elements in G that map to the identity in H
  • Provides information about the "collapse" of elements under the homomorphism
  • Always a normal subgroup of G
• Image of a homomorphism is the set of all elements in H that are mapped to by elements in G
  • Subset of the codomain group H
  • Isomorphic to the quotient group G/Ker(f)

Isomorphisms

• Bijective homomorphisms that preserve group structure completely
• Allow identification of groups with the same structure but different representations
• Satisfy $f(ab) = f(a)f(b)$ and have an inverse function that is also a homomorphism
• Crucial for group classification and understanding structural equivalence

Cosets and quotient groups

Left and right cosets

• Partitions of a group G with respect to a subgroup H
• Left coset: $aH = \{ah : h \in H\}$ for $a \in G$
• Right coset: $Ha = \{ha : h \in H\}$ for $a \in G$
• Provide insight into the structure of the group relative to its subgroups

Lagrange's theorem

• States that the order of a subgroup H of a finite group G divides the order of G
• |G| = |H| * [G:H], where [G:H] is the index of H in G (number of cosets)
• Imposes restrictions on possible subgroup sizes
• Fundamental result with wide-ranging implications in group theory

Normal subgroups

• Subgroups N of G where left and right cosets coincide: aN = Na for all a in G
• Equivalently, conjugation by any group element maps N to itself
• Allow for the construction of quotient groups
• Play a crucial role in group homomorphisms and group decomposition

Quotient groups

• Formed by taking the set of all cosets of a normal subgroup N in G
• Denoted as G/N, read as "G mod N"
• Elements are cosets, with the group operation defined on representatives
• Provide a way to create new groups from existing ones, crucial in classification theory

Group actions

Definition of group actions

• Describe how a group G acts on a set X through a function $\phi: G \times X \rightarrow X$
• Must satisfy identity and compatibility properties
• Formalize the concept of symmetry in mathematics
• Connect group theory to other areas of mathematics and science

Orbits and stabilizers

• Orbit of an element x in X is the set of all elements that can be reached by applying group elements to x
  • Partitions the set X into equivalence classes
• Stabilizer of x is the subgroup of G that leaves x fixed
  • Provides information about the symmetries of specific elements
• Orbit-Stabilizer Theorem relates the size of an orbit to the index of the stabilizer in G

Burnside's lemma

• Also known as the Cauchy-Frobenius lemma
• Counts the number of orbits in a group action
• Stated as: $|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|$, where $X^g$ is the set of elements fixed by g
• Useful in combinatorics and enumeration problems involving symmetry

Applications of group theory

Symmetry in mathematics

• Describes and classifies symmetries in geometric objects and abstract structures
• Fundamental in crystallography for understanding crystal structures
• Applied in physics to study conservation laws and fundamental particles
• Enables the classification of regular polyhedra and tilings

Cryptography

• Utilized in the design of secure communication protocols
• Elliptic curve cryptography relies heavily on group theory concepts
• Public key cryptosystems often based on the difficulty of certain group-theoretic problems
• Group-based cryptographic primitives offer potential quantum-resistant alternatives

Molecular structure

• Helps predict and analyze molecular geometries and bonding
• Point groups classify molecules based on their symmetry elements
• Facilitates the understanding of spectroscopic properties and chemical reactivity
• Applied in computational chemistry for efficient molecular calculations