definition 4.73 External semidirect product
open in the book ·
parts/02-mathematical-methods/02-algebraic-structures.tex:2682
· p. 90
- ground object -- no derivation owed
Rests on
- depends_on definition 4.1 Cartesian product ¶
-
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 4.3 Algebraic structure ¶
-
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 4.6 Associativity ¶
- depends_on definition 4.4 Internal binary operation; magma ¶
-
depends_on
definition 4.11
Inverse element
¶
-
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 3.23
Predicate
¶
- depends_on definition 3.28 Set ¶ ↺
-
depends_on
definition 3.24
Quantifiers
¶
-
depends_on
definition 4.21
Group
¶
-
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 4.44
Group isomorphism
¶
- depends_on definition 4.36 Subgroup ¶
-
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 4.45
Group automorphism
¶
Supports
-
depends_on
example 4.78
The Euclidean group
¶
-
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- depends_on remark 18.22 Superseded, and in exactly what sense ¶
-
depends_on
corollary 18.24
Galilean composition of velocities
¶
-
depends_on
proposition 18.20
The Galilean group
¶
- depends_on example 4.79 The Poincaré group ¶
-
depends_on
proposition 4.74
The semidirect product is a group
¶
-
depends_on
proposition 4.75
The two factors inside the semidirect product
¶
- depends_on example 4.79 The Poincaré group ¶ ↺
-
depends_on
proposition 4.77
When a semidirect product is direct
¶
- depends_on example 4.78 The Euclidean group ¶ ↺
-
depends_on
proposition 4.75
The two factors inside the semidirect product
¶
- depends_on proposition 18.20 The Galilean group ¶ ↺
- depends_on theorem 4.76 Internal characterization of the semidirect product ¶
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 |
→ | Cartesian product | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2692 |
depends_on |
→ | Group homomorphism | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2692 |
depends_on |
→ | The automorphism group | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2692 |
depends_on |
← | The Euclidean group | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2985 |
depends_on |
← | The Poincaré group | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:3015 |
depends_on |
← | The semidirect product is a group | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2709 |
depends_on |
← | The Galilean group | declared | parts/03-classical-mechanics/01-kinematics.tex:575 |
depends_on |
← | Internal characterization of the semidirect product | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2848 |