🔢Lower Division Math Foundations Unit 9 – Algebraic Structures: Groups & Fields

Key Concepts and Definitions

• Group consists of a set and a binary operation that satisfies certain axioms (closure, associativity, identity, and inverses)
• Field extends the concept of a group by adding additional axioms for multiplication and division
  • Includes a set, addition operation, and multiplication operation
• Binary operation combines two elements from a set to produce a unique element within the same set
• Identity element leaves other elements unchanged when the binary operation is applied
  • Additive identity is typically denoted as 0, while multiplicative identity is denoted as 1
• Inverse element "undoes" the effect of the binary operation
  • Additive inverse of $a$ is $-a$, while multiplicative inverse of $a$ is $a^{-1}$ (provided $a \neq 0$)
• Abelian group has a commutative binary operation, meaning the order of the operands does not affect the result
• Cyclic group can be generated by a single element through repeated application of the binary operation

Axioms and Properties

• Closure axiom states that for any elements $a$ and $b$ in a set $G$, the result of the binary operation $a * b$ is also in $G$
• Associativity axiom $(a * b) * c = a * (b * c)$ for all elements $a$, $b$, and $c$ in the set
• Identity axiom guarantees the existence of an identity element $e$ such that $a * e = e * a = a$ for all elements $a$ in the set
• Inverse axiom ensures that for each element $a$ in the set, there exists an inverse element $a^{-1}$ such that $a * a^{-1} = a^{-1} * a = e$
• Commutativity property $a * b = b * a$ holds for Abelian groups but not necessarily for all groups
• Cancellation property allows for "canceling out" common factors on both sides of an equation in a group
  • If $a * c = b * c$, then $a = b$ (left cancellation)
  • If $c * a = c * b$, then $a = b$ (right cancellation)

Types of Groups

• Finite groups have a finite number of elements in their set
  • Examples include the symmetric group $S_n$ and the cyclic group $\mathbb{Z}_n$
• Infinite groups have an infinite number of elements in their set
  • Examples include the integers under addition $(\mathbb{Z}, +)$ and the nonzero real numbers under multiplication $(\mathbb{R} \setminus \{0\}, \times)$
• Symmetric group $S_n$ consists of all permutations of $n$ distinct objects under the composition operation
• Dihedral group $D_n$ represents the symmetries of a regular $n$-gon, including rotations and reflections
• General linear group $GL(n, \mathbb{R})$ contains all invertible $n \times n$ matrices with real entries under matrix multiplication
• Special linear group $SL(n, \mathbb{R})$ is a subgroup of $GL(n, \mathbb{R})$ consisting of matrices with determinant 1

Group Operations and Examples

• Addition modulo $n$ forms a group $(\mathbb{Z}_n, +)$ with elements $\{0, 1, \ldots, n-1\}$
  • Example: $(\mathbb{Z}_5, +)$ has elements $\{0, 1, 2, 3, 4\}$ and addition is performed modulo 5
• Multiplication modulo $n$ forms a group $(\mathbb{Z}_n^*, \times)$ with elements that are coprime to $n$
  • Example: $(\mathbb{Z}_7^*, \times)$ has elements $\{1, 2, 3, 4, 5, 6\}$ and multiplication is performed modulo 7
• Matrix multiplication forms a group with the set of invertible matrices
  • Example: $GL(2, \mathbb{R})$ contains all invertible $2 \times 2$ matrices with real entries
• Function composition forms a group with the set of bijective functions from a set to itself
  • Example: The set of all permutations of $\{1, 2, 3\}$ under function composition is isomorphic to $S_3$

Subgroups and Cyclic Groups

• Subgroup is a subset of a group that forms a group under the same binary operation
  • Must satisfy the closure, associativity, identity, and inverse axioms
• Cyclic subgroup is generated by a single element through repeated application of the binary operation
  • Example: $\{0, 3, 6\}$ is a cyclic subgroup of $(\mathbb{Z}_9, +)$ generated by 3
• Lagrange's theorem states that the order (size) of a subgroup divides the order of the parent group
• Cosets are obtained by shifting a subgroup by a fixed element
  • Left coset of $H$ by $a$ is $aH = \{ah : h \in H\}$
  • Right coset of $H$ by $a$ is $Ha = \{ha : h \in H\}$
• Normal subgroup has the property that its left and right cosets coincide
  • Allows for the construction of quotient groups

Fields and Their Characteristics

• Field $(\mathbb{F}, +, \times)$ consists of a set $\mathbb{F}$ with addition and multiplication operations satisfying axioms
  • Additive group $(\mathbb{F}, +)$ is Abelian
  • Multiplicative group $(\mathbb{F} \setminus \{0\}, \times)$ is Abelian
  • Distributive property holds: $a \times (b + c) = (a \times b) + (a \times c)$
• Characteristic of a field is the smallest positive integer $n$ such that $\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0$
  • If no such $n$ exists, the field has characteristic 0
• Finite fields (Galois fields) have prime power order $p^n$ and are denoted as $GF(p^n)$
  • Example: $GF(4)$ has elements $\{0, 1, \alpha, \alpha^2\}$ where $\alpha^2 = \alpha + 1$
• Algebraic closure of a field $\mathbb{F}$ is the smallest field containing $\mathbb{F}$ in which every polynomial with coefficients in $\mathbb{F}$ has a root

Homomorphisms and Isomorphisms

• Homomorphism is a function $\phi$ between two groups $(G, *)$ and $(H, \diamond)$ that preserves the group structure
  • $\phi(a * b) = \phi(a) \diamond \phi(b)$ for all $a, b \in G$
• Isomorphism is a bijective homomorphism, establishing an equivalence between two groups
  • Isomorphic groups have the same structure and properties, differing only in the notation of their elements
• Kernel of a homomorphism $\phi : G \to H$ is the set of elements in $G$ that map to the identity in $H$
  • $\ker(\phi) = \{g \in G : \phi(g) = e_H\}$, where $e_H$ is the identity element in $H$
• First Isomorphism Theorem states that for a homomorphism $\phi : G \to H$, the quotient group $G / \ker(\phi)$ is isomorphic to the image of $\phi$ in $H$

Applications and Real-World Examples

• Cryptography utilizes the properties of finite fields and cyclic groups
  • Diffie-Hellman key exchange relies on the difficulty of the discrete logarithm problem in cyclic groups
  • Elliptic curve cryptography uses the group structure of points on an elliptic curve over a finite field
• Symmetry groups describe the symmetries of objects and patterns
  • The symmetries of a square form the dihedral group $D_4$, which includes rotations and reflections
  • The symmetries of a molecule can be represented by a subgroup of the symmetric group $S_n$
• Rubik's Cube and other permutation puzzles can be analyzed using group theory
  • The set of all possible configurations of a Rubik's Cube forms a group under the operation of applying a sequence of moves
• Quantum mechanics employs the unitary group $U(n)$ and the special unitary group $SU(n)$
  • Unitary matrices represent the transformations of quantum states
• Error-correcting codes, such as the Reed-Solomon code, use finite fields to detect and correct errors in data transmission and storage