proposition 18.20 The Galilean group

open in the book · parts/03-classical-mechanics/01-kinematics.tex:550 · p. 725

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 definition 18.18 Galilean transformation ¶
- depends_on definition 18.8 Inertial frame of reference ¶
  - depends_on definition 18.7 Frame of reference ¶
    - depends_on definition 18.3 Particle ¶
  - depends_on definition 18.6 Free particle ¶
    - depends_on definition 18.3 Particle ¶ ↺
- depends_on postulate 18.14 Absolute time ¶
- depends_on postulate 18.16 Principle of Galilean relativity ¶
depends_on example 4.78 The Euclidean group ¶
- depends_on definition 4.73 External semidirect product ¶ ↺
- 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 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 ¶
proves proof ch:01-kinematics@proof-4 ¶

Supports

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

Edges

type	direction	node	provenance	where
`depends_on`	→	External semidirect product	declared	parts/03-classical-mechanics/01-kinematics.tex:575
`depends_on`	→	Galilean transformation	declared	parts/03-classical-mechanics/01-kinematics.tex:575
`depends_on`	→	The Euclidean group	declared	parts/03-classical-mechanics/01-kinematics.tex:575
`depends_on`	←	Galilean composition of velocities	declared	parts/03-classical-mechanics/01-kinematics.tex:742
`depends_on`	←	Superseded, and in exactly what sense	declared	parts/03-classical-mechanics/01-kinematics.tex:684
`proves`	←	ch:01-kinematics@proof-4	declared	parts/03-classical-mechanics/01-kinematics.tex:579