11.1 Group Theory Fundamentals and Representations

Group theory is a powerful tool in mathematical physics, providing a framework to understand symmetries and transformations. It starts with basic properties like closure and associativity, then builds up to more complex concepts like subgroups and homomorphisms.

From simple multiplication tables to the classification of groups, this topic lays the groundwork for deeper analysis. Understanding these fundamentals is crucial for tackling advanced problems in quantum mechanics, particle physics, and other areas of mathematical physics.

Group Theory Fundamentals

Basic properties of groups

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• Set $G$ with binary operation $*$ forms a group if it satisfies closure, associativity, identity, and inverse properties
• Closure: $a, b \in G$ implies $a * b \in G$ (combining any two elements yields another element in the group)
• Associativity: $(a * b) * c = a * (b * c)$ for all $a, b, c \in G$ (order of applying the operation doesn't matter)
• Identity: Exists $e \in G$ such that $a * e = e * a = a$ for all $a \in G$ (element that leaves others unchanged under the operation)
• Inverses: For each $a \in G$, exists $a^{-1} \in G$ satisfying $a * a^{-1} = a^{-1} * a = e$ (element that "undoes" the operation)

Group multiplication tables

• Square table showing results of group operation for each pair of elements in a finite group
• Elements listed along top row and left column, with $i$-th row and $j$-th column entry representing result of operation on $i$-th and $j$-th elements
• Each element appears exactly once in each row and column
• Identity element appears along main diagonal
• Inverse of an element found in same row or column as identity element

Classification of groups

• Abelian groups: Commutative under group operation, i.e., $a * b = b * a$ for all $a, b \in G$ (examples: $(\mathbb{Z}, +)$, $(\mathbb{R} \setminus \{0\}, \cdot)$)
• Cyclic groups: Generated by a single element $a \in G$, with every element expressible as a power of $a$ (examples: $(\mathbb{Z}_n, +_n)$, rotational symmetries of a regular polygon)
• Symmetric groups: $S_n$, group of all permutations of $n$ distinct objects under composition, with order $n!$ (non-abelian for $n \geq 3$)

Group Theory Concepts

Subgroups, cosets and factor groups

• Subgroups: Subset $H \subseteq G$ forming a group under the same operation as $G$
  • Must contain identity element of $G$
  • Closure under operation and inverses within $H$
• Cosets: For subgroup $H \leq G$ and $a \in G$, left coset $aH = \{ah : h \in H\}$ and right coset $Ha = \{ha : h \in H\}$
  • Cosets partition $G$ into disjoint subsets
  • In abelian groups, left and right cosets coincide
• Factor groups (quotient groups): For normal subgroup $N \trianglelefteq G$, $G/N = \{aN : a \in G\}$ with operation $(aN)(bN) = (ab)N$

Homomorphisms and isomorphisms in groups

• Homomorphisms: Function $\phi: G \to H$ preserving group operation, i.e., $\phi(a * b) = \phi(a) * \phi(b)$ for all $a, b \in G$
  • Kernel $\ker(\phi) = \{a \in G : \phi(a) = e_H\}$, where $e_H$ is identity in $H$
• Isomorphisms: Bijective homomorphism, with $G \cong H$ denoting isomorphic groups
• Fundamental theorem of homomorphisms: For homomorphism $\phi: G \to H$, exists unique isomorphism $\bar{\phi}: G/\ker(\phi) \to \operatorname{im}(\phi)$ such that $\phi = \iota \circ \bar{\phi} \circ \pi$, where:
  1. $\pi: G \to G/\ker(\phi)$ is natural projection
  2. $\iota: \operatorname{im}(\phi) \to H$ is inclusion map