example 4.78 The Euclidean group

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2934 · p. 93

example -- no derivation owed by its kind

Rests on

depends_on definition 4.73 External semidirect product ¶
- depends_on definition 4.1 Cartesian product ¶
  - depends_on definition 3.28 Set ¶
- depends_on definition 4.43 Group homomorphism ¶
  - depends_on definition 4.21 Group ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶ ↺
- depends_on proposition 4.72 The automorphism group ¶
  - depends_on definition 4.45 Group automorphism ¶
    - depends_on definition 4.44 Group isomorphism ¶
      - depends_on definition 4.43 Group homomorphism ¶ ↺
      - depends_on definition 3.47 Bijective map ¶
  - depends_on definition 4.36 Subgroup ¶
    - depends_on definition 4.21 Group ¶ ↺
  - depends_on definition 4.48 Symmetric group ¶
    - depends_on definition 3.47 Bijective map ¶ ↺
  - proves proof ch:02-algebraic-structures@proof-27 ¶
depends_on proposition 4.77 When a semidirect product is direct ¶
- depends_on definition 4.65 Direct product of groups ¶
  - depends_on definition 4.1 Cartesian product ¶ ↺
  - depends_on definition 4.21 Group ¶ ↺
- depends_on definition 4.60 Normal subgroup ¶
  - depends_on definition 4.59 Conjugate subgroups ¶
    - depends_on definition 4.58 Conjugacy classes ¶
      - depends_on definition 3.59 Equivalence class ¶
      - depends_on proposition 4.57 prop:alg-conjugacy-equiv ¶
    - depends_on definition 4.36 Subgroup ¶ ↺
  - depends_on definition 4.36 Subgroup ¶ ↺
- depends_on proposition 4.75 The two factors inside the semidirect product ¶
  - depends_on definition 4.60 Normal subgroup ¶ ↺
  - depends_on definition 4.67 Product of subgroups ¶
    - depends_on definition 4.36 Subgroup ¶ ↺
  - depends_on proposition 4.74 The semidirect product is a group ¶
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.73 External semidirect product ¶ ↺
    - proves proof ch:02-algebraic-structures@proof-28 ¶
  - proves proof ch:02-algebraic-structures@proof-29 ¶
- proves proof ch:02-algebraic-structures@proof-31 ¶

Supports

depends_on proposition 18.20 The Galilean group ¶
- depends_on corollary 18.24 Galilean composition of velocities ¶
  - depends_on phenomenon 18.26 Composition of velocities, and its limit ¶
  - depends_on remark 18.25 Where the additivity goes ¶
- depends_on remark 18.22 Superseded, and in exactly what sense ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	External semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2985
`depends_on`	→	When a semidirect product is direct	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2985
`depends_on`	←	The Galilean group	declared	parts/03-classical-mechanics/01-kinematics.tex:575