proposition 4.68 Direct product of abelian groups

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:2444 · p. 88

Rests on

depends_on definition 4.8 Commutativity; abelian structure ¶
- depends_on definition 4.4 Internal binary operation; magma ¶
  - depends_on definition 4.3 Algebraic structure ¶
    - depends_on definition 4.2 Binary operation ¶
      - depends_on definition 4.1 Cartesian product ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 4.2 Binary operation ¶ ↺
depends_on definition 4.65 Direct product of groups ¶
- depends_on definition 4.1 Cartesian product ¶ ↺
- 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 ¶ ↺
proves proof ch:02-algebraic-structures@proof-25 ¶

Supports

depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶

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`	→	Commutativity; abelian structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2446
`depends_on`	→	Direct product of groups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2446
`depends_on`	←	The Cayley table of $\Z_2\times\Z_4$	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2488
`proves`	←	ch:02-algebraic-structures@proof-25	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2449