Abstract Algebra: Group Theory

Abstract Algebra

• Abstract algebra is a broad area of mathematics dealing with algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras
• It goes beyond the traditional algebra of solving equations to study the underlying structures themselves
• It's used across mathematics and physics, and it forms the basis for modern cryptography and coding theory

Group Theory

• A group is a set equipped with a binary operation that satisfies four fundamental axioms: closure, associativity, identity, and invertibility
• Closure means that for any two elements in the group, their combination under the operation is also within the group
• Associativity requires that the order of operations does not affect the result when combining three or more elements
• The identity axiom states that there exists a unique element in the group which, when combined with any other element, leaves the other element unchanged
• Invertibility means that each element in the group has a unique inverse element, such that their combination results in the identity element
• Groups can be finite or infinite, depending on the number of elements they contain
• The order of a group is the number of elements in the group
• Groups can be classified based on their properties, such as whether they are abelian (commutative), cyclic, or simple
• Group theory has applications in physics (symmetry in particle physics), chemistry (molecular symmetry), and computer science (cryptography).

Group

• A group is a set (G) with a binary operation ( * ) that combines any two elements (a) and (b) in (G) to form another element in (G), denoted as (a * b)
• The operation ( * ) must satisfy four axioms:
  • Closure: For all (a, b) in (G), (a * b) is also in (G)
  • Associativity: For all (a, b, c) in (G), ((a * b) * c = a * (b * c))
  • Identity: There exists an element (e) in (G) such that for all (a) in (G), (a * e = e * a = a). This element (e) is called the identity element
  • Inverse: For each (a) in (G), there exists an element (b) in (G) such that (a * b = b * a = e). This element (b) is called the inverse of (a), denoted as (a^{-1})
• A group (G) is called abelian (or commutative) if for all (a, b) in (G), (a * b = b * a)
• If a group is not abelian, it is called non-abelian
• The order of a group (G), denoted as (|G|), is the number of elements in (G). It can be finite or infinite
• A subgroup (H) of a group (G) is a subset of (G) that is itself a group under the same operation as (G)
• A cyclic group is a group that can be generated by a single element. That is, there exists an element (a) in (G) such that every element in (G) can be written as a power of (a)
• A permutation of a set (S) is a bijective (one-to-one and onto) function from (S) to itself
• The set of all permutations of a set (S) forms a group under the operation of function composition. This group is called the symmetric group on (S), denoted as (Sym(S)) or (S_n) if (S) has (n) elements
• Lagrange's Theorem states that if (G) is a finite group and (H) is a subgroup of (G), then the order of (H) divides the order of (G). That is, (|H|) divides (|G|)
• A normal subgroup (N) of a group (G) is a subgroup such that for all (g) in (G) and (n) in (N), (gng^{-1}) is in (N)
• If (N) is a normal subgroup of (G), then the quotient group (G/N) is the group whose elements are the cosets of (N) in (G), and the group operation is defined as ((aN)(bN) = (ab)N)
• A group homomorphism is a function between two groups that preserves the group structure
• Specifically, if (G) and (H) are groups, a function (\phi: G \to H) is a homomorphism if for all (a, b) in (G), (\phi(a * b) = \phi(a) * \phi(b))
• The kernel of a homomorphism (\phi: G \to H), denoted as (ker(\phi)), is the set of elements in (G) that map to the identity element in (H). That is, (ker(\phi) = {g \in G \mid \phi(g) = e_H}), where (e_H) is the identity element in (H)
• The image of a homomorphism (\phi: G \to H), denoted as (im(\phi)), is the set of elements in (H) that are the image of some element in (G). That is, (im(\phi) = {\phi(g) \mid g \in G})
• The First Isomorphism Theorem states that if (\phi: G \to H) is a group homomorphism, then (G/ker(\phi)) is isomorphic to (im(\phi)). In symbols, (G/ker(\phi) \cong im(\phi))

Assonance

• Assonance is the repetition of similar vowel sounds in words that are close to each other
• It often occurs in poetry and prose to create a musical effect, reinforce a mood, or draw attention to certain words
• It differs from rhyme in that the consonant sounds do not need to be the same, only the vowel sounds
• It differs from consonance, which involves repeated consonant sounds

Examples of Assonance

• "The cat sat back." (repetition of the "a" sound)
• "Men sell the wedding bells." (repetition of the "e" sound)
• "I lie down by the side of my bride" (repetition of the "i" sound)
• "Stony broke" (repetition of the "o" sound)
• "Try to light the fire" (repetition of the "i" sound)

Function of Assonance

• Enhances the musicality of language, making it more pleasing and memorable
• Creates internal rhyme and rhythm
• Connects words and ideas through sound
• Reinforces the meaning or mood of a passage
• Adds emphasis to specific words or phrases
• Can create a sense of unity or coherence in a text

Distinctive features of Assonance

• Repetition of vowel sounds
• Can occur at the beginning, middle, or end of words
• Does not require the repetition of consonant sounds
• Creates a subtle effect, often less obvious than rhyme
• Used in poetry, prose, and even everyday speech