🏃🏽‍♀️Galois Theory Unit 6 – Introduction to Group Theory

What's Group Theory All About?

• Branch of abstract algebra that studies the algebraic structures known as groups
• Groups are sets equipped with an operation that combines any two elements to form a third element in the set
• Group theory examines the axiomatic properties of these operations and elements
• Investigates concepts such as symmetry, permutations, and transformations
• Provides a foundation for understanding algebraic structures and their relationships
• Has applications in various fields including physics, chemistry, and computer science
• Allows for the classification and analysis of abstract objects based on their structural properties

Key Concepts and Definitions

• Group: A set $G$ together with a binary operation $*$ that satisfies the group axioms (closure, associativity, identity, and inverses)
• Abelian group: A group in which the binary operation is commutative, i.e., $a * b = b * a$ for all $a, b \in G$
• Order of a group: The number of elements in the group, denoted as $|G|$
• Identity element: An element $e \in G$ such that $a * e = e * a = a$ for all $a \in G$
• Inverse element: For each $a \in G$, there exists an element $b \in G$ such that $a * b = b * a = e$, where $e$ is the identity element
• Cyclic group: A group generated by a single element, i.e., all elements can be obtained by repeatedly applying the group operation to a specific element
• Subgroup: A subset $H$ of a group $G$ that is itself a group under the same operation as $G$

Types of Groups

• Finite groups: Groups with a finite number of elements (symmetric groups, dihedral groups, cyclic groups)
• Infinite groups: Groups with an infinite number of elements (integers under addition, real numbers under multiplication)
• Symmetric groups: The group of all permutations of a set, denoted as $S_n$ for a set with $n$ elements
• Dihedral groups: The group of symmetries of a regular polygon, including rotations and reflections
• Matrix groups: Groups of invertible matrices under matrix multiplication (general linear groups, orthogonal groups, special linear groups)
• Lie groups: Continuous groups that are also smooth manifolds, often used in physics and geometry
• Topological groups: Groups equipped with a topology such that the group operations are continuous

Group Operations and Properties

• Binary operation: A function that takes two elements of a set and produces a single element of the same set
• Closure: For all $a, b \in G$, the result of the operation $a * b$ is also in $G$
• Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
• Identity element: The unique element $e \in G$ such that $a * e = e * a = a$ for all $a \in G$
• Inverse elements: For each $a \in G$, there exists a unique element $b \in G$ such that $a * b = b * a = e$
• Commutativity: A group is commutative (or Abelian) if $a * b = b * a$ for all $a, b \in G$
  • Commutative groups have additional properties and simplifications compared to non-commutative groups
• Cancellation laws: For all $a, b, c \in G$, if $a * b = a * c$, then $b = c$ (left cancellation), and if $b * a = c * a$, then $b = c$ (right cancellation)

Subgroups and Cosets

• Subgroup: A non-empty subset $H$ of a group $G$ that is closed under the group operation and contains inverses
  • The identity element of $G$ must be in $H$
  • For all $a, b \in H$, $a * b \in H$
  • For each $a \in H$, $a^{-1} \in H$
• Coset: A subset of a group obtained by applying the group operation to a fixed element and all elements of a subgroup
  • Left coset: For $a \in G$ and a subgroup $H$, the left coset is $aH = \{a * h : h \in H\}$
  • Right coset: For $a \in G$ and a subgroup $H$, the right coset is $Ha = \{h * a : h \in H\}$
• Lagrange's Theorem: If $G$ is a finite group and $H$ is a subgroup of $G$, then the order of $H$ divides the order of $G$
  • The order of any element in a finite group divides the order of the group
• Normal subgroup: A subgroup $N$ of $G$ is normal if $aN = Na$ for all $a \in G$, i.e., left and right cosets coincide
  • Normal subgroups are essential for constructing quotient groups

Homomorphisms and Isomorphisms

• Group homomorphism: A function $\phi: G \to H$ between two groups that preserves the group operation, i.e., $\phi(a * b) = \phi(a) * \phi(b)$ for all $a, b \in G$
• Kernel of a homomorphism: The set of all elements in $G$ that map to the identity element of $H$ under the homomorphism $\phi$, i.e., $\ker(\phi) = \{a \in G : \phi(a) = e_H\}$
  • The kernel is always a normal subgroup of $G$
• Isomorphism: A bijective group homomorphism, i.e., a one-to-one correspondence between two groups that preserves the group operation
  • If there exists an isomorphism between two groups, they are essentially the same group up to relabeling of elements
• First Isomorphism Theorem: For a group homomorphism $\phi: G \to H$, there is a natural isomorphism between the image of $\phi$ and the quotient group $G / \ker(\phi)$
• Automorphism: An isomorphism from a group to itself, i.e., a bijective homomorphism $\phi: G \to G$
  • The set of all automorphisms of a group $G$ forms a group under function composition, denoted as $\text{Aut}(G)$

Applications and Examples

• Symmetry groups: Groups that describe the symmetries of geometric objects or physical systems (point groups, space groups, wallpaper groups)
• Rubik's Cube: The set of all possible configurations of a Rubik's Cube forms a group under the operation of applying a sequence of moves
• Cryptography: Group theory is used in various cryptographic algorithms, such as the Diffie-Hellman key exchange and elliptic curve cryptography
• Quantum mechanics: Groups are used to describe the symmetries of quantum systems and to classify elementary particles
• Coding theory: Group theory is applied in the design and analysis of error-correcting codes, such as linear codes and cyclic codes
• Combinatorics: Groups are used to study counting problems, such as the number of distinct ways to color a symmetric pattern
• Music theory: Group theory can be used to analyze the structure and symmetries of musical compositions

Common Pitfalls and Tips

• Forgetting to check all group axioms: When determining if a set with an operation forms a group, make sure to verify closure, associativity, the existence of an identity element, and the existence of inverses
• Confusing left and right cosets: Always pay attention to the order of the operation when dealing with cosets, as $aH$ and $Ha$ may be different for non-normal subgroups
• Misapplying Lagrange's Theorem: Remember that the converse of Lagrange's Theorem is not true; if $d$ divides the order of a group $G$, there may not necessarily be a subgroup of order $d$
• Overlooking the importance of normal subgroups: Normal subgroups are crucial for constructing quotient groups and for the study of group homomorphisms
• Not exploiting the properties of specific types of groups: When working with particular types of groups (e.g., cyclic, Abelian, or symmetric), make use of their additional properties to simplify problems
• Seeking connections with other mathematical structures: Group theory is closely related to other algebraic structures, such as rings and fields, as well as to topology and geometry; understanding these connections can provide valuable insights
• Practicing problem-solving: To develop a deep understanding of group theory, it is essential to work through a variety of problems and examples, ranging from simple to more challenging ones