definition 4.43 Group homomorphism

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:1481 · p. 75

ground object -- no derivation owed

Rests on

depends_on definition 4.21 Group ¶
- depends_on definition 4.11 Inverse element ¶
  - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
- depends_on definition 4.20 Monoid ¶
  - depends_on definition 4.9 Neutral element ¶ ↺
  - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
      - depends_on definition 4.2 Binary operation ¶ ↺
depends_on definition 3.43 Map ¶
- depends_on definition 3.24 Quantifiers ¶
  - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.1 Logical proposition ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on definition 3.28 Set ¶

Supports

depends_on definition 4.44 Group isomorphism ¶
- depends_on definition 4.45 Group automorphism ¶
  - depends_on proposition 4.72 The automorphism group ¶
    - depends_on definition 4.73 External semidirect product ¶
      - depends_on example 4.78 The Euclidean group ¶
      - depends_on example 4.79 The Poincaré group ¶
      - depends_on proposition 4.74 The semidirect product is a group ¶
      - depends_on proposition 18.20 The Galilean group ¶
      - depends_on theorem 4.76 Internal characterization of the semidirect product ¶
- depends_on definition 5.143 Faithful representation ¶
- depends_on proposition 4.70 Characterization of the direct product ¶
- depends_on proposition 4.55 $S_3$ and the equilateral triangle ¶
- depends_on theorem 4.56 Cayley's theorem ¶
depends_on definition 4.73 External semidirect product ¶ ↺
depends_on definition 5.141 Representation of a group ¶
- depends_on definition 5.146 Character ¶
- depends_on definition 5.142 Dimension of a representation ¶
- depends_on definition 5.145 Equivalent representations ¶
  - depends_on proposition 5.155 Averaging trick ¶
    - depends_on theorem 101.62 The mixing matrix is unitary ¶
      - depends_on proposition 101.63 Parameter counting, and why three generations permit $CP$ violation ¶
      - depends_on theorem 103.11 GIM suppression at one loop ¶
  - depends_on theorem 5.160 Schur's second lemma ¶
    - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on example 25.45 Weyl ordering collapses ¶
    - depends_on theorem 25.38 Groenewold–van Hove ¶
      - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
      - depends_on remark 24.55 What polarization costs ¶
- depends_on definition 5.143 Faithful representation ¶ ↺
- depends_on definition 5.147 Invariant subspace ¶
  - depends_on definition 5.148 Irreducible representation ¶
    - depends_on proposition 5.161 Irreducible representations of an abelian group ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
      - depends_on theorem 101.62 The mixing matrix is unitary ¶ ↺
    - depends_on theorem 5.158 Schur's first lemma ¶
      - depends_on corollary 14.18 A Casimir acts as a number on an irreducible representation ¶
      - depends_on lemma A.427 Invariant bilinear forms on a simple algebra ¶
      - depends_on lemma A.405 Schur, algebra form ¶
      - depends_on phenomenon 102.1 Hadron masses obey an octet mass formula ¶
      - depends_on proposition 5.161 Irreducible representations of an abelian group ¶ ↺
      - depends_on theorem 106.72 Confinement at strong coupling ¶
      - depends_on theorem 102.12 Which quark combinations can be colour singlets ¶
    - depends_on theorem 5.160 Schur's second lemma ¶ ↺
  - depends_on definition 5.149 Totally reducible representation ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
  - depends_on lemma 5.150 Invariance of the orthogonal complement ¶
    - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
    - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
  - depends_on theorem 5.158 Schur's first lemma ¶ ↺
  - depends_on theorem 5.160 Schur's second lemma ¶ ↺
- depends_on definition 5.144 Unitary representation ¶
  - depends_on lemma 5.150 Invariance of the orthogonal complement ¶ ↺
  - depends_on proposition 5.155 Averaging trick ¶ ↺
depends_on proposition 4.42 When the homomorphisms form a group ¶
depends_on proposition 4.47 Image of the inverse element ¶
depends_on proposition 4.46 Image of the neutral element ¶
- depends_on proposition 4.47 Image of the inverse element ¶ ↺
depends_on proposition 4.17 Trivial kernel of a homomorphism ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1487
`depends_on`	→	Map	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1487
`depends_on`	←	Group isomorphism	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1495
`depends_on`	←	External semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2692
`depends_on`	←	Representation of a group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6190
`depends_on`	←	When the homomorphisms form a group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1436
`depends_on`	←	Image of the inverse element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1618
`depends_on`	←	Image of the neutral element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1597
`depends_on`	←	Trivial kernel of a homomorphism	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:438