definition 4.65 Direct product of groups
open in the book ·
parts/02-mathematical-methods/02-algebraic-structures.tex:2381
· p. 87
- ground object -- no derivation owed
Rests on
- depends_on definition 4.1 Cartesian product ¶
-
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 4.3
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 4.3 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 4.6
Associativity
¶
-
depends_on
definition 4.11
Inverse element
¶
Supports
-
depends_on
proposition 4.68
Direct product of abelian groups
¶
- depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶
- depends_on proposition 4.70 Characterization of the direct product ¶
- depends_on proposition 4.66 prop:alg-direct-product-group ¶
-
depends_on
proposition 4.77
When a semidirect product is direct
¶
-
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- depends_on remark 18.22 Superseded, and in exactly what sense ¶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 4.78
The Euclidean group
¶
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:2393 |
depends_on |
→ | Group | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2393 |
depends_on |
← | Direct product of abelian groups | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2446 |
depends_on |
← | Characterization of the direct product | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2541 |
depends_on |
← | prop:alg-direct-product-group | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2398 |
depends_on |
← | When a semidirect product is direct | declared | parts/02-mathematical-methods/02-algebraic-structures.tex:2900 |