theorem 4.76 Internal characterization of the semidirect product

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2831 · p. 92

Rests on

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 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 4.48 Symmetric group ¶
    - depends_on definition 3.47 Bijective map ¶ ↺
  - proves proof ch:02-algebraic-structures@proof-27 ¶
depends_on definition 4.67 Product of subgroups ¶
- depends_on definition 4.36 Subgroup ¶ ↺
proves proof ch:02-algebraic-structures@proof-30 ¶

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type	direction	node	provenance	where
`depends_on`	→	Normal subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2848
`depends_on`	→	External semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2848
`depends_on`	→	Product of subgroups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2848
`proves`	←	ch:02-algebraic-structures@proof-30	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2852