2.3 Groups, Semigroups, and Monoids
5 min read•august 16, 2024
Algebraic structures like groups, semigroups, and monoids are key building blocks in abstract algebra. They're defined by specific rules that govern how elements interact, with groups having the most structure and semigroups the least.
These structures pop up everywhere, from number systems to matrices. Understanding their properties and how they relate to each other is crucial for tackling more complex algebraic concepts and solving real-world problems in math and beyond.
Groups, Semigroups, and Monoids
Definitions and Key Characteristics
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- consists of a set S and a • with associative property
- builds on semigroup by adding an e
- For all a in S, a • e = e • a = a
- extends monoid by including inverse elements
- Every element a in S has an inverse a^(-1) where a • a^(-1) = a^(-1) • a = e
- Algebraic strength hierarchy groups > monoids > semigroups
- Structures differ in presence of identity and inverse elements
- Real-world examples
- Group (integers under addition)
- Monoid (natural numbers under addition)
- Semigroup (natural numbers under multiplication)
Properties and Operations
- Semigroups require for all a, b, c in S: (a • b) • c = a • (b • c)
- Monoids add identity element to semigroup properties
- Groups incorporate four key properties
- ensures a • b remains in S for all a, b in S
- Associativity as in semigroups
- Identity element e exists where a • e = e • a = a for all a in S
- a^(-1) exists for every a, satisfying a • a^(-1) = a^(-1) • a = e
- Apply these properties to classify algebraic structures
- Analyze commutativity, order, and substructures in examples
Advanced Concepts and Theorems
- Cancellation law in groups states if a • b = a • c or b • a = c • a, then b = c
- Identity elements in monoids and groups are unique
- Inverse elements in groups are unique
- Group properties include (a^(-1))^(-1) = a and (a • b)^(-1) = b^(-1) • a^(-1)
- connects element order to group order in finite groups
- Prove theorems using axiom application and logical reasoning
- formed by bijective functions on set X under composition
Axioms of Algebraic Structures
Fundamental Axioms
- Associativity forms the foundation for semigroups, monoids, and groups
- (a • b) • c = a • (b • c) for all a, b, c in set S
- Closure ensures operation results remain within the set
- a • b is in S for all a, b in S
- Identity element e exists in monoids and groups
- a • e = e • a = a for all a in S
- Inverse elements a^(-1) exist for each element a in groups
- a • a^(-1) = a^(-1) • a = e
Axiom Applications
- Verify semigroup structure by checking associativity
- Confirm monoid structure by validating associativity and identity element
- Establish group structure by ensuring all four axioms hold
- Apply axioms to prove properties like cancellation law in groups
- Use axioms to derive additional theorems (uniqueness of identity and inverses)
- Analyze algebraic structures in abstract algebra and number theory
Axiom Implications
- Associativity allows flexible grouping in calculations
- Closure ensures operations stay within the defined set
- Identity element provides a neutral element for the operation
- Inverse elements enable "undoing" operations
- Group axioms support more advanced algebraic manipulations
- Axioms form the basis for studying homomorphisms and isomorphisms
- Understanding axioms aids in constructing and analyzing new algebraic structures
Examples of Algebraic Structures
Number Systems
- Integers (Z) under addition form a group
- Identity: 0
- Inverse: negation
- Natural numbers (N) under addition create a monoid
- Identity: 0
- No inverses for positive integers
- Positive integers (Z+) under multiplication form a monoid
- Identity: 1
- No inverses except for 1
- Rational numbers (Q) without 0 under multiplication form a group
- Identity: 1
- Inverse: reciprocal
Matrix and Function Structures
- n x n matrices under matrix multiplication create a monoid
- Identity: identity matrix
- Invertible n x n matrices under multiplication form a group (general linear group)
- Bijective functions on set X under composition form symmetric group
- Continuous functions on interval [0,1] under addition form a group
- Polynomial functions under addition form a group
Abstract Algebraic Structures
- Cyclic groups generated by a single element
- Dihedral groups representing symmetries of regular polygons
- Quaternion group with eight elements
- Free groups generated by a set of symbols
- Abelian groups with commutative operation
Theorems for Algebraic Structures
Fundamental Theorems
- Cancellation law in groups
- If a • b = a • c or b • a = c • a, then b = c
- Uniqueness of identity elements in monoids and groups
- Uniqueness of inverse elements in groups
- Inverse of inverse equals the original element (a^(-1))^(-1) = a
- Inverse of product equals product of inverses in reverse order (a • b)^(-1) = b^(-1) • a^(-1)
- Lagrange's theorem states element order divides group order in finite groups
Proof Techniques
- Direct proofs using axiom application and logical deduction
- Proof by contradiction for uniqueness theorems
- Induction for theorems involving repeated operations
- Counterexamples to disprove false statements about algebraic structures
- Diagram chasing for more complex algebraic relationships
Applications and Extensions
- Use cancellation law to simplify equations in groups
- Apply uniqueness theorems to identify specific elements in structures
- Utilize inverse properties in solving group equations
- Employ Lagrange's theorem in analyzing structure
- Extend theorems to study subgroups and quotient groups
- Apply theorems in cryptography and coding theory (group-based algorithms)
- Use algebraic structure theorems in abstract algebra research
Key Terms to Review (19)
Abelian group: An abelian group is a set equipped with a binary operation that combines any two elements to form a third element, satisfying four key properties: closure, associativity, identity, and the existence of inverses, along with the additional requirement that the operation is commutative. This means that the order in which you combine elements does not affect the outcome. Abelian groups are fundamental in various areas of algebra, especially when examining structures and transformations, and they serve as building blocks for more complex mathematical concepts.
Associativity: Associativity is a fundamental property of certain binary operations that states the way in which operations are grouped does not affect the result. This means that for any three elements, the operation can be performed in any order of grouping without changing the outcome, typically expressed as $$(a * b) * c = a * (b * c)$$.
Binary operation: A binary operation is a mathematical function that combines two elements from a set to produce another element from the same set. This operation takes two inputs, typically denoted as 'a' and 'b', and produces an output, denoted as 'a * b'. Binary operations form the foundation for various algebraic structures and are essential for understanding concepts such as groups, semigroups, and monoids.
Cayley's Theorem: Cayley's Theorem states that every group is isomorphic to a subgroup of the symmetric group acting on its elements. This theorem highlights the connection between groups and permutations, showing that any abstract group can be represented as a group of symmetries. The importance of this theorem lies in its ability to provide a concrete realization of abstract algebraic structures through the lens of permutation groups, which are more intuitive and easier to visualize.
Closure: Closure refers to the property of a set where the result of applying a specific operation on elements of that set always yields another element that is also within the same set. This concept is crucial in understanding how operations behave within algebraic structures, ensuring that performing operations does not lead to results outside the defined set, which maintains the integrity of the structure.
Closure Property: The closure property refers to the idea that a set is closed under a specific operation if applying that operation to any elements within the set results in an element that is also within the same set. This concept is crucial for understanding the structure and behavior of algebraic systems, as it helps define the characteristics of various structures such as groups, semigroups, and algebras.
Cyclic Group: A cyclic group is a type of group that can be generated by a single element, known as a generator. This means that every element of the group can be expressed as a power (or multiple) of this generator. Cyclic groups can be either finite or infinite, and they exhibit a simple structure that makes them an essential concept in group theory, linking to other fundamental concepts like homomorphisms and subgroup properties.
Dihedral Group: The dihedral group is a mathematical structure that represents the symmetries of a regular polygon, including both rotations and reflections. It captures the idea of how many ways you can move a shape around while keeping its appearance unchanged, making it a fundamental example of a finite group in abstract algebra. The dihedral group of order $n$ is denoted as $D_n$ and consists of $n$ rotations and $n$ reflections, highlighting its dual nature as both a rotational and reflectional symmetry group.
Finite Group: A finite group is a set equipped with a binary operation that satisfies the group properties of closure, associativity, identity, and invertibility, with the additional condition that the set contains a finite number of elements. Finite groups play a crucial role in abstract algebra, allowing for the classification and analysis of algebraic structures. Understanding finite groups is fundamental to studying their homomorphic images, subgroups, and direct product structures, making them a cornerstone in the study of algebraic systems.
Group: A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverse elements for every element in the set. Understanding groups is crucial as they serve as foundational structures in algebra, enabling us to analyze symmetries and transformations in various mathematical contexts.
Group homomorphism: A group homomorphism is a function between two groups that preserves the group operation, meaning that the image of the product of two elements in the first group equals the product of their images in the second group. This concept connects different algebraic structures by providing a way to map one group onto another while maintaining the inherent structure of both groups. Understanding homomorphisms is crucial for exploring more advanced topics such as isomorphisms, which are specific types of homomorphisms that are also bijective.
Identity Element: An identity element is a special type of element in a set with a binary operation that, when combined with any element of the set, returns that same element. This concept is essential because it establishes a foundational behavior in structures like groups, semigroups, and monoids, ensuring that there exists an element that acts neutrally with respect to the operation involved. Understanding identity elements helps in grasping more complex algebraic structures and their properties.
Inverse element: An inverse element is an element in a set that, when combined with another specific element (usually referred to as the identity element), produces the identity element itself. This concept is crucial in understanding how operations can reverse each other's effects, allowing for the solution of equations and manipulation of algebraic structures. Inverse elements help define groups and other algebraic systems where certain conditions must be satisfied for operations to hold consistently.
Isomorphism: Isomorphism is a mathematical concept that describes a structural similarity between two algebraic structures, meaning there is a one-to-one correspondence between their elements that preserves operations. This concept is crucial for understanding how different structures can be considered equivalent in terms of their algebraic properties, regardless of their specific representations or contexts.
Lagrange's Theorem: Lagrange's Theorem states that in a finite group, the order of any subgroup divides the order of the group itself. This fundamental result in group theory connects the sizes of groups and their subgroups, showing how the structure of a group is related to its subgroups. It emphasizes the importance of subgroup orders and lays the groundwork for further exploration of group properties and classifications.
Monoid: A monoid is an algebraic structure consisting of a set equipped with a binary operation that is associative and has an identity element. This means that for any three elements in the set, the operation can be performed in any grouping without changing the outcome, and there is a specific element in the set that, when used with any other element in the operation, returns that same element. Monoids are essential building blocks in abstract algebra, connecting closely with groups and semigroups by emphasizing their structure and properties.
Non-empty set: A non-empty set is a collection of elements that contains at least one member. This concept is crucial in algebraic structures, as the existence of at least one element often determines the behavior and properties of mathematical operations and relations defined within that set.
Semigroup: A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation. This means that if you take any three elements from the set and apply the operation to them in any grouping, the result will be the same. Semigroups serve as a foundational concept in abstract algebra, connecting to groups and monoids, as they share the associative property but do not necessarily require an identity element or inverses.
Symmetric group: The symmetric group is the group consisting of all possible permutations of a finite set. It plays a crucial role in the study of algebraic structures, as it encapsulates the concept of symmetry and transformations, connecting deeply with groups, semigroups, and monoids through the idea of operations on sets.