proposition 4.74 The semidirect product is a group

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2702 · p. 91

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 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 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 ¶
proves proof ch:02-algebraic-structures@proof-28 ¶

Supports

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 18.20 The Galilean group ¶
      - depends_on corollary 18.24 Galilean composition of velocities ¶
      - depends_on remark 18.22 Superseded, and in exactly what sense ¶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`	→	Group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2709
`depends_on`	→	External semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2709
`depends_on`	←	The two factors inside the semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2786
`proves`	←	ch:02-algebraic-structures@proof-28	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2712