proposition 4.42 When the homomorphisms form a group

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:1429 · p. 75

Rests on

depends_on definition 4.43 Group homomorphism ¶
- 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 3.43 Map ¶
  - depends_on definition 3.24 Quantifiers ¶
    - depends_on definition 3.23 Predicate ¶
      - depends_on definition 3.1 Logical proposition ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on definition 3.28 Set ¶
depends_on definition 4.36 Subgroup ¶
- depends_on definition 4.21 Group ¶ ↺
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 ¶
depends_on proposition 4.14 Maps into a group form a group ¶
- depends_on definition 4.21 Group ¶ ↺
- depends_on equation 4.6 eq:alg-opposite-map ¶
- depends_on proposition 4.23 Uniqueness of the inverse element ¶ ↺
- proves proof ch:02-algebraic-structures@proof-2 ¶
proves proof ch:02-algebraic-structures@proof-12 ¶

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type	direction	node	provenance	where
`depends_on`	→	Group homomorphism	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1436
`depends_on`	→	Subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1436
`depends_on`	→	Inverse of a product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1436
`depends_on`	→	Maps into a group form a group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1436
`proves`	←	ch:02-algebraic-structures@proof-12	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1439