definition 4.9 Neutral element

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:180 · p. 59

ground object -- no derivation owed

Rests on

depends_on definition 4.3 Algebraic structure ¶
- depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.1 Cartesian product ¶
    - depends_on definition 3.28 Set ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶ ↺

Supports

depends_on definition 4.29 Action ¶
- depends_on definition 4.31 Module ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.34 Algebra ¶
    - depends_on definition 5.121 Algebra ¶
      - depends_on definition 5.128 Derivation ¶
      - depends_on definition 5.124 Ideal ¶
      - depends_on definition 5.127 Lie algebra ¶
      - depends_on definition 5.122 Linear hull ¶
      - depends_on definition 5.123 Subalgebra ¶
      - depends_on definition 5.162 Representation of an algebra ¶
      - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
    - depends_on definition 5.101 External direct sum ¶
      - depends_on definition 12.84 External direct sum ¶
      - depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
      - depends_on proposition 5.102 Dimensions add ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on corollary 5.29 Orthogonal decomposition ¶
      - depends_on definition 7.96 Plane ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 12.94 Tensor product of Hilbert spaces ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 5.24 Orthogonal vectors ¶
      - depends_on definition 5.144 Unitary representation ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on example 12.11 The function space $L^{2}$ ¶
      - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - depends_on proposition 5.63 prop:lin-dual-inner-product ¶
      - … 5 more
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.101 External direct sum ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.7 Vector subspace ¶
      - depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶
      - depends_on proposition 5.107 The union of two subspaces ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 12.69 Operator with a domain ¶
      - depends_on definition 5.41 Adjoint ¶ ↺
      - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.128 Derivation ¶ ↺
      - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.46 Functional ¶
      - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.53 Isomorphism ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.66 Pushforward ¶
      - … 6 more
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.22 Metric associated with a norm ¶
      - depends_on definition 5.25 Unit vector ¶
      - depends_on definition 9.21 Matrix exponential ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
    - depends_on definition 5.3 Scalar ¶
    - depends_on definition 5.115 Sesquilinear form ¶
    - depends_on definition 5.4 Vector ¶
    - depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶ ↺
    - depends_on example 5.10 Functions on a set ¶
    - … 1 more
depends_on definition 4.11 Inverse element ¶
- depends_on definition 4.32 Field ¶
  - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on lemma A.44 $\R$ is an ordered field ¶
  - depends_on theorem A.35 Completeness of $\R$ ¶
- depends_on definition 4.21 Group ¶
  - depends_on definition 4.37 Cayley table ¶
    - depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶
  - depends_on definition 4.65 Direct product of groups ¶
    - depends_on proposition 4.68 Direct product of abelian groups ¶
      - depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶ ↺
    - depends_on proposition 4.70 Characterization of the direct product ¶
    - depends_on proposition 4.66 prop:alg-direct-product-group ¶
    - depends_on proposition 4.77 When a semidirect product is direct ¶
      - depends_on example 4.78 The Euclidean group ¶
  - depends_on definition 4.35 The group axioms, restated one at a time ¶
  - depends_on definition 4.43 Group homomorphism ¶
    - depends_on definition 4.44 Group isomorphism ¶
      - depends_on definition 4.45 Group automorphism ¶
      - depends_on definition 5.143 Faithful representation ¶
      - depends_on proposition 4.70 Characterization of the direct product ¶ ↺
      - depends_on proposition 4.55 $S_3$ and the equilateral triangle ¶
      - depends_on theorem 4.56 Cayley's theorem ¶
    - 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 18.20 The Galilean group ¶
      - depends_on theorem 4.76 Internal characterization of the semidirect product ¶
    - depends_on definition 5.141 Representation of a group ¶
      - depends_on definition 5.146 Character ¶
      - depends_on definition 5.142 Dimension of a representation ¶
      - depends_on definition 5.145 Equivalent representations ¶
      - depends_on definition 5.143 Faithful representation ¶ ↺
      - depends_on definition 5.147 Invariant subspace ¶
      - depends_on definition 5.144 Unitary representation ¶ ↺
    - depends_on proposition 4.42 When the homomorphisms form a group ¶
    - depends_on proposition 4.47 Image of the inverse element ¶
    - depends_on proposition 4.46 Image of the neutral element ¶
      - depends_on proposition 4.47 Image of the inverse element ¶ ↺
    - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶
  - depends_on definition 4.31 Module ¶ ↺
  - depends_on definition 4.49 Permutation group ¶
    - depends_on proposition 4.50 Cardinal of the permutation group ¶
      - depends_on proposition 4.52 $S_3$ is a smallest non-abelian group ¶
    - depends_on proposition 4.53 The subgroups of $S_3$ ¶
    - depends_on theorem 4.56 Cayley's theorem ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.32 Field ¶ ↺
    - depends_on definition 4.31 Module ¶ ↺
  - depends_on definition 4.36 Subgroup ¶
    - depends_on definition 4.59 Conjugate subgroups ¶
      - depends_on definition 4.60 Normal subgroup ¶
    - 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 proposition 4.61 prop:alg-coset-equiv ¶
      - depends_on definition 4.62 Cosets ¶
    - 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 ¶ ↺
  - depends_on definition A.561 Coadjoint action and equivariance ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
    - depends_on theorem A.563 Marsden–Weinstein reduction ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
      - depends_on example A.572 Rotational reduction of the central-force problem ¶
  - depends_on definition 5.141 Representation of a group ¶ ↺
  - depends_on example 4.22 A two-element group ¶
  - depends_on proposition 4.40 prop:alg-congruence-equiv ¶
    - depends_on proposition 4.41 prop:alg-zn-group ¶
      - depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶ ↺
  - … 11 more
- depends_on proposition 4.17 Trivial kernel of a homomorphism ¶ ↺
depends_on definition 4.15 Kernel ¶
- depends_on proposition 4.17 Trivial kernel of a homomorphism ¶ ↺
depends_on definition 4.20 Monoid ¶
- depends_on definition 4.32 Field ¶ ↺
- depends_on definition 4.21 Group ¶ ↺
depends_on proposition 4.10 Uniqueness of the neutral element ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Algebraic structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:187
`depends_on`	←	Action	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:757
`depends_on`	←	Inverse element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:228
`depends_on`	←	Kernel	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:413
`depends_on`	←	Monoid	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:537
`depends_on`	←	Uniqueness of the neutral element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:199