proposition 4.52 $S_3$ is a smallest non-abelian group

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:1815 · p. 79

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 4.21 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 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
depends_on proposition 4.50 Cardinal of the permutation group ¶
- depends_on definition 4.49 Permutation group ¶
  - depends_on definition 4.21 Group ¶ ↺
  - depends_on definition 4.48 Symmetric group ¶
    - depends_on definition 3.47 Bijective map ¶
- proves proof ch:02-algebraic-structures@proof-15 ¶
proves proof ch:02-algebraic-structures@proof-16 ¶

Supports

Nothing declares a dependency on this node yet.

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:1819
`depends_on`	→	Group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1819
`depends_on`	→	Cardinal of the permutation group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1819
`proves`	←	ch:02-algebraic-structures@proof-16	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1822