definition 4.59 Conjugate subgroups

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2065 · p. 83

ground object -- no derivation owed

Rests on

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 4.11 Inverse element ¶
      - 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 3.58 Equivalence relation ¶ ↺
  - depends_on proposition 4.27 Inverse of a product ¶
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on proposition 4.23 Uniqueness of the inverse element ¶
      - depends_on definition 4.21 Group ¶ ↺
      - proves proof ch:02-algebraic-structures@proof-4 ¶
    - proves proof ch:02-algebraic-structures@proof-8 ¶
  - proves proof ch:02-algebraic-structures@proof-20 ¶
depends_on definition 4.36 Subgroup ¶
- depends_on definition 4.21 Group ¶ ↺

Supports

depends_on definition 4.60 Normal subgroup ¶
- depends_on proposition 4.63 Well-definedness of the coset product ¶
- depends_on proposition 4.70 Characterization of the direct product ¶
- 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 proposition 4.77 When a semidirect product is direct ¶ ↺
- depends_on theorem 4.64 Quotient 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`	→	Conjugacy classes	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2077
`depends_on`	→	Subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2077
`depends_on`	←	Normal subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2089