definition 4.60 Normal subgroup

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2080 · p. 83

ground object -- no derivation owed

Rests on

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 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
    - 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 ¶
      - 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 ¶ ↺
depends_on definition 4.36 Subgroup ¶ ↺

Supports

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 corollary 18.24 Galilean composition of velocities ¶
      - depends_on remark 18.22 Superseded, and in exactly what sense ¶
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`	→	Conjugate subgroups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2089
`depends_on`	→	Subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2089
`depends_on`	←	Well-definedness of the coset product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2277
`depends_on`	←	Characterization of the direct product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2541
`depends_on`	←	The two factors inside the semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2786
`depends_on`	←	When a semidirect product is direct	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2900
`depends_on`	←	Quotient group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2328
`depends_on`	←	Internal characterization of the semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2848