🔶Intro to Abstract Math Unit 3 – Algebraic Structures

Key Concepts and Definitions

• Algebraic structures are mathematical objects that consist of a set and one or more operations defined on the set
• Binary operation combines two elements of a set to produce another element in the same set
• Identity element leaves other elements unchanged when combined with the operation (e.g., 0 for addition, 1 for multiplication)
  • Additive identity is the element that when added to any element $a$ results in $a$
  • Multiplicative identity is the element that when multiplied by any element $a$ results in $a$
• Inverse element "undoes" the effect of the original element when combined using the operation
  • Additive inverse of an element $a$ is the element $-a$ such that $a + (-a) = 0$
  • Multiplicative inverse of a non-zero element $a$ is the element $a^{-1}$ such that $a \cdot a^{-1} = 1$
• Associativity is a property where the order of operations does not affect the result: $(a * b) * c = a * (b * c)$
• Commutativity is a property where the order of elements does not affect the result: $a * b = b * a$

Types of Algebraic Structures

• Groups are algebraic structures with a single binary operation that is associative, has an identity element, and every element has an inverse
  • Abelian groups are groups where the binary operation is also commutative
  • Examples of groups include integers under addition $(\mathbb{Z}, +)$, non-zero real numbers under multiplication $(\mathbb{R} \setminus \{0\}, \cdot)$
• Rings are algebraic structures with two binary operations (usually addition and multiplication) that satisfy certain axioms
  • Addition in a ring is an abelian group
  • Multiplication is associative and distributive over addition
  • Examples of rings include integers $(\mathbb{Z}, +, \cdot)$, polynomials with real coefficients $(\mathbb{R}[x], +, \cdot)$
• Fields are rings where the non-zero elements form an abelian group under multiplication
  • Examples of fields include rational numbers $(\mathbb{Q}, +, \cdot)$, real numbers $(\mathbb{R}, +, \cdot)$, complex numbers $(\mathbb{C}, +, \cdot)$
• Vector spaces are algebraic structures that consist of a set of vectors, a field of scalars, and operations of vector addition and scalar multiplication satisfying certain axioms
  • Examples of vector spaces include $\mathbb{R}^n$ over the field of real numbers, the set of all polynomials with real coefficients over the field of real numbers

Properties and Axioms

• Closure property ensures that the result of an operation on elements of a set is also an element of the same set
  • For a set $S$ and a binary operation $*$, if $a, b \in S$, then $a * b \in S$
• Associativity: $(a * b) * c = a * (b * c)$ for all elements $a, b, c$ in the set
• Commutativity: $a * b = b * a$ for all elements $a, b$ in the set
• Identity element $e$: $a * e = e * a = a$ for all elements $a$ in the set
• Inverse element: For each element $a$, there exists an element $b$ such that $a * b = b * a = e$
• Distributive property (for rings and fields): $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ for all elements $a, b, c$ in the set
• Axioms are fundamental properties that define an algebraic structure and must be satisfied by all elements in the set

Examples and Applications

• Symmetry groups in geometry describe the symmetries of objects (e.g., rotations, reflections)
  • Dihedral groups $D_n$ represent the symmetries of a regular n-gon
  • Symmetric groups $S_n$ represent the permutations of $n$ distinct objects
• Cryptography uses algebraic structures to secure communication and data
  • RSA encryption relies on the difficulty of factoring large integers, which is related to the multiplicative group of integers modulo $n$
• Physics uses algebraic structures to model physical phenomena
  • Lorentz group describes symmetries in special relativity
  • Lie groups and Lie algebras are used in quantum mechanics and particle physics
• Error-correcting codes use algebraic structures to detect and correct errors in data transmission
  • Linear codes, such as Hamming codes, use vector spaces over finite fields
• Computer graphics uses algebraic structures to represent transformations and manipulate objects in 2D and 3D space
  • Affine transformations (e.g., translations, rotations, scaling) form a group under composition

Homomorphisms and Isomorphisms

• Homomorphism is a function between two algebraic structures that preserves the operations
  • If $f: A \to B$ is a homomorphism, then $f(a * b) = f(a) \star f(b)$, where $*$ and $\star$ are the operations in $A$ and $B$, respectively
• Isomorphism is a bijective homomorphism, meaning it has an inverse that is also a homomorphism
  • If $f: A \to B$ is an isomorphism, then $A$ and $B$ are said to be isomorphic, denoted as $A \cong B$
  • Isomorphic structures have the same algebraic properties and are essentially the same, up to relabeling of elements
• Kernel of a homomorphism $f: A \to B$ is the set of elements in $A$ that map to the identity element in $B$: $\ker(f) = \{a \in A : f(a) = e_B\}$
• First Isomorphism Theorem states that if $f: A \to B$ is a homomorphism, then $A / \ker(f) \cong \operatorname{im}(f)$, where $\operatorname{im}(f)$ is the image of $f$ in $B$

Substructures and Quotients

• Substructure is a subset of an algebraic structure that is closed under the operations and forms the same type of structure
  • Subgroups are subsets of a group that form a group under the same operation
  • Subrings are subsets of a ring that form a ring under the same operations
  • Subspaces are subsets of a vector space that form a vector space under the same operations
• Cosets are subsets of a group obtained by shifting a subgroup by an element
  • Left coset of a subgroup $H$ by an element $a$ is the set $aH = \{ah : h \in H\}$
  • Right coset of a subgroup $H$ by an element $a$ is the set $Ha = \{ha : h \in H\}$
• Quotient structures are obtained by partitioning an algebraic structure using an equivalence relation
  • Quotient group $G/N$ is the set of cosets of a normal subgroup $N$ in $G$, which forms a group under coset multiplication
  • Quotient ring $R/I$ is the set of cosets of an ideal $I$ in $R$, which forms a ring under coset addition and multiplication
  • Quotient space $V/W$ is the set of cosets of a subspace $W$ in $V$, which forms a vector space under coset addition and scalar multiplication

Theorems and Proofs

• Lagrange's Theorem states that for a finite group $G$ and a subgroup $H$, the order of $H$ divides the order of $G$
  • Consequence: the order of an element divides the order of the group
• Cayley's Theorem states that every group is isomorphic to a subgroup of a symmetric group
  • Proves the existence of a faithful action of a group on a set
• Fundamental Theorem of Finite Abelian Groups states that every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order
  • Provides a classification of finite abelian groups up to isomorphism
• Fermat's Little Theorem states that if $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1} \equiv 1 \pmod{p}$
  • Useful in number theory and cryptography
• Proofs in abstract algebra often involve:
  • Direct manipulation of definitions and axioms
  • Constructing isomorphisms or homomorphisms
  • Induction on the size or structure of the algebraic objects
  • Contradiction, assuming the negation of the statement and deriving a false conclusion

Problem-Solving Techniques

• Identify the algebraic structure and its properties based on the given information
  • Determine if the set and operations form a group, ring, field, or vector space
  • Check if the structure is abelian, cyclic, or has other special properties
• Use definitions and axioms to prove or disprove statements
  • Verify that the axioms hold for a specific example or counterexample
  • Prove that a subset is a substructure by showing it is closed under the operations and satisfies the axioms
• Construct homomorphisms or isomorphisms to relate different structures
  • Define a function and prove that it preserves the operations
  • Use isomorphisms to transfer properties between structures
• Apply theorems to simplify problems or derive new results
  • Use Lagrange's Theorem to determine the possible orders of subgroups or elements
  • Use the First Isomorphism Theorem to relate quotient structures to homomorphic images
• Break down complex problems into smaller, more manageable parts
  • Prove properties separately for each operation in a ring or vector space
  • Decompose a group into simpler subgroups or quotient groups
• Look for patterns or analogies with familiar structures or problems
  • Recognize that a problem is similar to a well-known example or theorem
  • Adapt the solution or proof technique from a related problem