definition 4.21 Group

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:550 · p. 63

ground object -- no derivation owed

Rests on

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 4.1 Cartesian product ¶
      - depends_on definition 3.43 Map ¶
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 4.37 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 proposition 18.20 The Galilean group ¶
      - depends_on corollary 18.24 Galilean composition of velocities ¶
      - depends_on remark 18.22 Superseded, and in exactly what sense ¶
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 proposition 4.72 The automorphism group ¶
      - depends_on definition 4.73 External semidirect product ¶
  - 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 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 proposition 5.155 Averaging trick ¶
      - depends_on theorem 101.62 The mixing matrix is unitary ¶
    - depends_on theorem 5.160 Schur's second lemma ¶
      - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on theorem 25.38 Groenewold–van Hove ¶
  - depends_on definition 5.143 Faithful representation ¶ ↺
  - depends_on definition 5.147 Invariant subspace ¶
    - depends_on definition 5.148 Irreducible representation ¶
      - depends_on proposition 5.161 Irreducible representations of an abelian group ¶
      - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
      - depends_on theorem 5.158 Schur's first lemma ¶
      - depends_on theorem 5.160 Schur's second lemma ¶ ↺
    - depends_on definition 5.149 Totally reducible representation ¶
      - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
    - depends_on lemma 5.150 Invariance of the orthogonal complement ¶
      - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
    - depends_on theorem 5.158 Schur's first lemma ¶ ↺
    - depends_on theorem 5.160 Schur's second lemma ¶ ↺
  - depends_on definition 5.144 Unitary representation ¶
    - depends_on lemma 5.150 Invariance of the orthogonal complement ¶ ↺
    - depends_on proposition 5.155 Averaging trick ¶ ↺
- 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 4.33 Vector space ¶
  - depends_on definition 4.34 Algebra ¶
  - depends_on definition 5.121 Algebra ¶
    - depends_on definition 5.128 Derivation ¶
      - depends_on remark 22.31 The bracket is a Lie algebra structure ¶
    - depends_on definition 5.124 Ideal ¶
      - depends_on proposition 5.164 The kernel is an ideal ¶
    - depends_on definition 5.127 Lie algebra ¶
      - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on proposition 5.167 Adjoint representation ¶
      - depends_on proposition 5.139 Dimension of the symplectic group ¶
      - depends_on remark 22.31 The bracket is a Lie algebra structure ¶ ↺
    - depends_on definition 5.122 Linear hull ¶
      - depends_on definition 5.124 Ideal ¶ ↺
      - depends_on definition 5.123 Subalgebra ¶
    - depends_on definition 5.123 Subalgebra ¶ ↺
    - depends_on definition 5.162 Representation of an algebra ¶
      - depends_on definition 5.163 Faithful, equivalent, invariant ¶
      - depends_on proposition 5.167 Adjoint representation ¶ ↺
      - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
    - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
      - depends_on definition 5.162 Representation of an algebra ¶ ↺
      - depends_on definition 9.21 Matrix exponential ¶
      - depends_on proposition 5.139 Dimension of the symplectic group ¶ ↺
  - depends_on definition 5.101 External direct sum ¶
    - depends_on definition 12.84 External direct sum ¶
      - depends_on proposition 12.85 The direct sum is a Hilbert space ¶
    - depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
      - depends_on corollary 5.108 The sum is the right operation ¶
      - depends_on proposition 5.104 Criterion for a direct sum ¶
      - depends_on proposition 5.105 Grassmann's formula ¶
    - depends_on proposition 5.102 Dimensions add ¶
      - depends_on proposition 5.104 Criterion for a direct sum ¶ ↺
  - depends_on definition 5.18 Inner product ¶
    - depends_on corollary 5.29 Orthogonal decomposition ¶
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
      - depends_on theorem 5.43 The four fundamental subspaces ¶
    - depends_on definition 7.96 Plane ¶
    - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on definition 12.32 Separable Hilbert space ¶
      - depends_on definition 12.94 Tensor product of Hilbert spaces ¶
      - depends_on definition 10.85 Sobolev space ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on example 12.11 The function space $L^{2}$ ¶
      - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
      - depends_on theorem 16.70 The direct method ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - … 1 more
    - depends_on definition 12.94 Tensor product of Hilbert spaces ¶ ↺
    - depends_on definition 5.41 Adjoint ¶
      - depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on theorem 5.43 The four fundamental subspaces ¶ ↺
      - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
      - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
    - depends_on definition 5.24 Orthogonal vectors ¶
      - depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on definition 12.16 Orthogonal complement ¶
      - depends_on definition 5.26 Orthogonal basis ¶
    - 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 theorem 12.14 Closest point in a closed convex set ¶ ↺
    - 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 corollary 5.108 The sum is the right operation ¶ ↺
      - depends_on definition 5.15 Basis ¶
      - depends_on lemma 5.38 Exchange and completion ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on corollary 5.62 A functional that annihilates a set of constraints ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on lemma 5.38 Exchange and completion ¶ ↺
      - depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
      - depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
      - depends_on proposition 5.31 Gram criterion ¶
      - depends_on proposition 5.28 Gram–Schmidt ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on corollary 5.29 Orthogonal decomposition ¶ ↺
      - depends_on definition 12.13 Convex set ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.103 Sum of subspaces; internal direct sum ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.163 Faithful, equivalent, invariant ¶ ↺
      - depends_on definition 5.147 Invariant subspace ¶ ↺
      - depends_on example 5.10 Functions on a set ¶
      - depends_on proposition 5.61 Annihilator of a subspace ¶
      - depends_on proposition 5.107 The union of two subspaces ¶
    - 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 7.102 Differential of a function ¶
      - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on definition 12.88 Reducing subspace ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on proposition 12.36 Boundedness is continuity ¶
      - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
    - depends_on definition 12.69 Operator with a domain ¶
      - depends_on definition 12.70 Graph; closed and closable operators ¶
      - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶
    - depends_on definition 5.41 Adjoint ¶ ↺
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 12.94 Tensor product of Hilbert spaces ¶ ↺
      - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on lemma 5.137 The alternating top form is unique up to scale ¶
      - depends_on proposition 5.114 prop:lin-bilinear-dual ¶
      - depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      - depends_on theorem 5.116 Sylvester's law of inertia ¶
    - depends_on definition 5.128 Derivation ¶ ↺
    - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.57 Automorphism ¶
      - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.76 Diagonalizable operator ¶
      - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
    - depends_on definition 5.46 Functional ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
      - depends_on definition 5.59 Dual space ¶
      - depends_on definition 5.60 Functional specified on a basis ¶
    - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
    - depends_on definition 5.53 Isomorphism ¶
      - depends_on definition 5.57 Automorphism ¶ ↺
      - depends_on proposition 5.104 Criterion for a direct sum ¶ ↺
      - depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶
      - depends_on theorem 5.160 Schur's second lemma ¶ ↺
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
    - depends_on definition 5.66 Pushforward ¶
      - depends_on proposition 5.67 Adjointness of pullback and 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 5.27 Orthonormal basis ¶
    - depends_on definition 9.21 Matrix exponential ¶ ↺
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      - depends_on theorem 12.12 Riesz–Fischer ¶
  - 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 4.49 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 4.33 Vector space ¶ ↺
  - depends_on lemma A.44 $\R$ is an ordered field ¶
  - depends_on theorem A.35 Completeness of $\R$ ¶
- depends_on definition 4.31 Module ¶ ↺
depends_on definition 4.36 Subgroup ¶
- 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 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 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 ¶ ↺
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.570 Existence and uniqueness of the reduced form ¶
      - depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
- 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Inverse element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:563
`depends_on`	→	Monoid	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:563
`depends_on`	←	Cayley table	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1053
`depends_on`	←	Direct product of groups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2393
`depends_on`	←	The group axioms, restated one at a time	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:980
`depends_on`	←	Group homomorphism	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1487
`depends_on`	←	Module	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:814
`depends_on`	←	Permutation group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1667
`depends_on`	←	Ring	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:773
`depends_on`	←	Subgroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1017
`depends_on`	←	Coadjoint action and equivariance	declared	appendices/A-long-proofs.tex:27083
`depends_on`	←	Representation of a group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6190
`depends_on`	←	A two-element group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:591
`depends_on`	←	prop:alg-congruence-equiv	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1280
`depends_on`	←	prop:alg-conjugacy-equiv	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2015
`depends_on`	←	prop:alg-direct-product-group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2398
`depends_on`	←	Inverse of a product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:682
`depends_on`	←	Uniqueness of the inverse element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:604
`depends_on`	←	Maps into a group form a group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:361
`depends_on`	←	Inverse of the neutral element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:623
`depends_on`	←	Neutral element equal to a product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:643
`depends_on`	←	$S_3$ is a smallest non-abelian group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1819
`depends_on`	←	The semidirect product is a group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2709
`depends_on`	←	prop:alg-zn-group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1359
`depends_on`	←	Quoted: discrete subgroups of a real vector space	declared	appendices/A-long-proofs.tex:26689