definition 4.36 Subgroup

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:1004 · p. 69

ground object -- no derivation owed

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 ¶ ↺

Supports

depends_on definition 4.59 Conjugate subgroups ¶
- 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 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 ¶
depends_on definition 4.60 Normal subgroup ¶ ↺
depends_on definition 4.67 Product of subgroups ¶
- depends_on proposition 4.70 Characterization of the direct product ¶ ↺
- depends_on proposition 4.75 The two factors inside the semidirect product ¶ ↺
- depends_on theorem 4.76 Internal characterization of the semidirect product ¶ ↺
depends_on proposition 4.72 The automorphism group ¶
- depends_on definition 4.73 External semidirect product ¶
  - depends_on example 4.78 The Euclidean group ¶ ↺
  - depends_on example 4.79 The Poincaré group ¶ ↺
  - depends_on proposition 4.74 The semidirect product is a group ¶
    - depends_on proposition 4.75 The two factors inside the semidirect product ¶ ↺
  - 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 ¶
  - depends_on theorem 4.76 Internal characterization of the semidirect product ¶ ↺
depends_on proposition 4.61 prop:alg-coset-equiv ¶
- depends_on definition 4.62 Cosets ¶
  - depends_on proposition 4.63 Well-definedness of the coset product ¶ ↺
depends_on proposition 4.42 When the homomorphisms form a group ¶
depends_on proposition 4.38 prop:alg-nZ-subgroup ¶
depends_on proposition 4.53 The subgroups of $S_3$ ¶
depends_on theorem 4.56 Cayley's theorem ¶

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:1017
`depends_on`	←	Conjugate subgroups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2077
`depends_on`	←	Normal subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2089
`depends_on`	←	Product of subgroups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2441
`depends_on`	←	The automorphism group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2656
`depends_on`	←	prop:alg-coset-equiv	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2108
`depends_on`	←	When the homomorphisms form a group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1436
`depends_on`	←	prop:alg-nZ-subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1237
`depends_on`	←	The subgroups of $S_3$	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1861
`depends_on`	←	Cayley's theorem	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1948