proposition 4.77 When a semidirect product is direct

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2889 · p. 93

Rests on

depends_on definition 4.65 Direct product of groups ¶
- depends_on definition 4.1 Cartesian product ¶
  - depends_on definition 3.28 Set ¶
- 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 4.20 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 4.60 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 definition 3.58 Equivalence relation ¶
    - depends_on proposition 4.57 prop:alg-conjugacy-equiv ¶
      - depends_on definition 4.21 Group ¶ ↺
      - depends_on definition 3.58 Equivalence relation ¶ ↺
      - depends_on proposition 4.27 Inverse of a product ¶
      - proves proof ch:02-algebraic-structures@proof-20 ¶
  - depends_on definition 4.36 Subgroup ¶
    - depends_on definition 4.21 Group ¶ ↺
- 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 ¶
    - depends_on definition 4.1 Cartesian product ¶ ↺
    - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.21 Group ¶ ↺
      - depends_on definition 3.43 Map ¶
    - depends_on proposition 4.72 The automorphism group ¶
      - depends_on definition 4.45 Group automorphism ¶
      - depends_on definition 4.36 Subgroup ¶ ↺
      - depends_on definition 4.48 Symmetric group ¶
      - proves proof ch:02-algebraic-structures@proof-27 ¶
  - 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 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 ¶

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`	→	Direct product of groups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2900
`depends_on`	→	Normal subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2900
`depends_on`	→	The two factors inside the semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2900
`depends_on`	←	The Euclidean group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2985
`proves`	←	ch:02-algebraic-structures@proof-31	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2904