Index of terminology

Abelian group: I, § 4, no. 2. Absolute value of a rational number: I, § 9, no. 4. Action of one set on another: I, § 3, no. 1. Action, canonical: I, § 3, no. 1. Action, distributive: I, § 3, no. 4. Action, induced: I, § 3, no. 2. Action, quotient: I, § 3, no. 3. Action, right, left, derived from a law of composition: I, § 3, no. 1. Actions which commute: I, § 5, no. 4. Addition: I, § 1, no. 1. Addition of rational integers: I, § 2, no. 5. Additively (law written): I, § 1, no. 1. Adjoint of a matrix: III, § 11, Exercise 9. Adjunction of a unit element (algebra derived by): III, § 1, no. 2. Affine function: II, § 9, no. 4. Affine group: II, § 9, no. 4. Affine line, plane, hyperplane: II, § 9, no. 3. Affine mapping: II, § 9, no. 4. Affine space: II, § 9, no. 1. Affine subset, linear variety: II, § 9, no. 3. Affinely free, affinely related family: II, § 9, no. 3. Affinely independent points: II, § 9, no. 3. Algebra, A-Algebra: III, § 1, no. 1. Algebra, alternative: III, Appendix, no. 1. Algebra, alternating graded: III, § 4, no. 9. Algebra, anticommutative graded: III, § 4, no. 9. Algebra, associative: III, § 1, no. 1. Algebra, Cayley: III, § 2, no. 4. Algebra, commutative: III, § 1, no. 1.

Algebra derived from an algebra by the adjunction of a unit element: III, § 1, no. 2. Algebra, exterior, of a module: III, § 7, no. 1. Algebra, free: III, § 2, no. 7. Algebra, free associative: III, § 2, no. 7. Algebra, free associative and commutative: III, § 2, no. 7. Algebra, free, of a module: III, § 2, Exercise 13. Algebra generated by generators subject to relations: III, § 2, no. 8. Algebra, graded, over a graded ring: III, § 3, no. 1. Algebra, total, of a monoid: III, § 2, no. 10. Algebra obtained by extension of scalars: III, § 1, no. 5. Algebra obtained by restriction of scalars: III, § 1, no. 5. Algebra of a group, of a magma, of a monoid: III, § 2, no. 6. Algebra of dual numbers: III, § 2, no. 3. Algebra of formal power series: III, § 2, no. 11. Algebra of Hamiltonian quaternions: III, § 2, no. 5. Algebra of octonians of type $(\alpha, \beta, \gamma, \delta)$: III, Appendix, no. 3. Algebra of quaternions: III, § 2, no. 5. Algebra of quaternions of type $(\alpha, \beta, \gamma)$, of type $(\alpha, \gamma)$: III, § 2, no. 5. Algebra, opposite: III, § 1, no. 1. Algebra, product: III, § 1, no. 4. Algebra, quadratic: III, § 2, no. 3. Algebra, quadratic, of type $(\alpha, \beta)$: III, § 2, no. 3. Algebra, quotient: III, § 1, no. 4. Algebra, Rees: III, § 6, Exercise 4. Algebra, restricted, of a monoid: III, § 2, no. 10. Algebra, symmetric, of a module: III, § 6, no. 1. Algebra, tensor, of a module: III, § 5, no. 1. Algebra, unital: III, § 1, no. 1. Algebra, universal, defined by a generating system related by a family of relators: III, § 2, no. 8. Algebra, universal, generated by a set subjected to identities: III, § 2, no. 8. Algebra, universal unital associative, defined by a generating system related by a family of relators: III, § 2, no. 8. Algebras, linearly disjoint: III, § 4, no. 4. Alternating graded algebra: III, § 4, no. 9. Alternating group: III, § 5, no. 7. Alternating multilinear mapping: III, § 7, no. 4. Alternative algebra: III, Appendix, no. 1. Amalgamated sum: I, § 7, no. 3. Annihilated by a scalar (element): II, § 1, no. 12. Annihilator, left, right: I, § 8, no. 6. Annihilator, of a subset, of an element of a module: II, § 1, no. 12.

Antiautomorphism: III, § 1, no. 1. Anticocommutative graded cogebra: III, § 11, no. 3. Anticocommutative skew graded bigebra: III, § 11, no. 4. Anticommutative graded algebra: III, § 4, no. 9. Anticommutative skew graded bigebra: III, § 11, no. 4. Antiderivation, K-antiderivation: III, § 10, no. 2. Antiendomorphism of a ring: II, § 10, no. 6. Associated (B-module) with an A-module and a ring homomorphism B → A: II, § 1, no. 13. Associated (faithful module) with a module: II, § 1, no. 12. Associated (law of action) with an action: I, § 3, no. 1. Associated (linear mapping) with an affine linear mapping: II, § 9, no. 4. Associated (vectorspace) with a module over an integral domain: II, § 7, no. 10. Associated (vector subspace) with a homogeneous element of an exterior algebra: III, § 7, no. 2. Associated (vector subspace) with a homogeneous element of a symmetric algebra: III, § 6, no. 2. Associated (vector subspace) with a homogeneous element of a tensor algebra: III, § 5, no. 2. Associative algebra: III, § 1, no. 1. Associative algebra, free: III, § 2, no. 7. Associative and commutative algebra, free: III, § 2, no. 7. Associative law: I, § 1, no. 3. Associativity relations in a multiplication table: III, § 1, no. 7. Associativity theorem: I, § 1, no. 3. Associator: III, Appendix, no. 1. Attached (affine space) to a vector space: II, § 9, no. 3. Augmentation: III, § 10, no. 8. Automorphism, inner of a group: I, § 5, no. 3. Automorphism, inner, of a ring: I, § 8, no. 4. Automorphism with no fixed point: I, § 6, Exercise 23.

Barycentre of m points, barycentre of a family of weighted points: II, § 9, no. 2. Barycentric coordinate: II, § 9, no. 3. Bases dual to one another: II, § 2, no. 7. Basic family in a group: I, § 7, no. 6. Basis dual of a basis of a module: II, § 2, no. 6. Basis, Hamel: II, § 7, no. 1. Basis of a module: II, § 1, no. 11. Basis of an algebra: III, § 1, no. 7. Basis of T_I^J(M) associated with a basis of M: III, § 5, no. 6.

Basis of type $(\alpha, \beta)$ of a quadratic algebra: III, § 2, no. 3. Basis of type $(\alpha, \beta, \gamma)$, of type $(\alpha, \gamma)$, of a quaternion algebra: III, § 2, no. 5. Basis, projective: II, § 9, Exercise 10. Biadditive, $\mathbf{Z}$-bilinear mapping: II, § 3, no. 1. Bicentralizer: I, § 1, no. 5. Bicentralizer of a subalgebra: III, § 1, no. 2. Bicyclic group: I, § 6, Exercise 26. Bidual of a module: II, § 2, no. 7. Bigebra, anticommutative, anticocommutative, skew graded: III, § 11, no. 4 Bigebra, cocommutative, commutative, graded: III, § 11, no. 4. Bigebra, graded bigebra, skew graded bigebra, III, § 11, no. 4. Bigebra of a monoid: III, § 11, no. 4. Bigraded group, ring, module: II, § 11, no. 2. Bigraduation: II, § 11, no. 1. Bilinear mapping: II, § 3, no. 5. Bimodule, (A, B)-bimodule: II, § 1, no. 14. Bimodule over algebras: III, § 4, no. 3. Binomial formula: I, § 8, no. 2. Block product of matrices: II, § 10, no. 5. Boolean ring: I, § 9, Exercise 8. Bordered matrix: II, § 10, no. 1. Bracket, $\varepsilon$-bracket of two derivations: III, § 10, no. 4.

Cancellable, left, right, cancellable, element: I, § 2, no 2. Cartan-Brauer-Hua Theorem: I, § 9, Exercise 18. Cayley algebra: III, § 2, no. 4. Cayley extension of an algebra: III, § 2, no. 5. Cayley-Hamilton theorem: III, § 8, no. 11. Cayley norm, trace: III, § 2, no. 4. Cayley octonians: III, Appendix, no. 3. Central element: I, § 1, no. 5. Central extension: I, § 6, no. 1. Central homothety: II, § 1, no. 2. Central ring homomorphism: II, § 5, no. 3. Central series, lower: I, § 6, no. 3. Centralizer: I, § 5, no. 3. Centralizer of a subalgebra of an associative algebra: III, § 1, no. 2. Centralizer of a subset: I, § 1, no. 5. Centralizer of a subset of a field: II, § 7, no. 7. Centralizer subalgebra: III, § 1, no. 2. Centralizing element: I, § 5, no. 3. Centralizing subset: I, § 5, no. 3. Centre: I, § 1, no. 5.

Centre of an algebra: III, § 1, no. 2. Centre of a projective linear mapping: II, § 9, no. 10. Change of coordinates, formulae of: II, § 10, no. 8. Characteristic of a field: I, § 9, Exercise 4. Characteristic polynomial of a matrix: III, § 8, no. 11. Characteristic subgroup: I, § 5, no. 3. Class, conjugacy: I, § 5, no. 4. Class, nilpotency, of a group: I, § 6, no. 3. Class, solvability, of a group: I, § 6, no. 4. Coassociative cogebra: III, § 11, no. 2. Cocommutative bigebra: III, § 11, no. 4. Cocommutative cogebra: III, § 11, no. 2. Codiagonal mapping: II, § 1, no. 6. Codimension of an affine linear variety: II, § 9, no. 3. Codimension of a vector subspace: II, § 7, no. 3. Coefficients of a formal power series: III, § 2, no. 11. Coefficients of a linear combination: II, § 1, no. 1. Coefficients of a polynomial: III, § 2, no. 9. Coefficients of a system of linear equations: II, § 2, no. 8. Cofactor of an element of a square matrix: III, § 8, no. 6. Cogebra, A-cogebra: III, § 11, no. 1. Cogebra, anticocommutative graded: III, § 11, no. 3. Cogebra, coassociative: III, § 11, no. 2. Cogebra, cocommutative: III, § 11, no. 2. Cogebra, counital: III, § 11, no. 2. Cogebra, graded: III, § 11, no. 1. Cogebra, opposite: III, § 11, no. 1. Coimage of a linear mapping: II, § 1, no. 3. Coincidence group: I, § 4, no. 8. Cokernel of a linear mapping: II, § 1, no. 3. Column of a matrix: II, § 10, no. 1. Combination, linear: II, § 2, no. 5. Combinations, formal linear (module of): II, § 1, no. 11. Commutation factor: III, § 10, no. 1. Commutative algebra: III, § 1, no. 1. Commutative field: I, § 9, no. 1. Commutative graded bigebra: III, § 11, no. 4. Commutative group, free, over X: I, § 7, no. 5. Commutative group with operators: I, § 4, no. 2. Commutative law: I, § 1, no. 5. Commutative magma: I, § 1, no. 5. Commutative monoid, free, over X: I, § 7, no. 7. Commutative ring: I, § 8, no. 1.

Commutativity relations in a multiplication table: III, § 1, no. 7. Commutativity theorem: I, § 1, no. 5. Commutator group: I, § 6, no. 2. Commutator of two elements: I, § 6, no. 2. Commute, actions which: I § 5, no. 4. Commute, elements which: I, § 1, no. 5. Compatible (equivalence relation) with a law of composition: I, § 1, no. 6. Compatible (equivalence relation) with an action: I, § 3, no. 3. Compatible (graduation) with a coproduct: III, § 11, no. 1. Compatible (graduation) with an algebra structure: III, § 3, no. 1. Compatible (graduation) with a ring, module, structure: II, § 11, no. 2. Compatible law of composition and equivalence relation: I, § 1, no. 6. Compatible, left, right (equivalence relation), with a law of composition: I, § 3, no. 3. Compatible (mapping) with an action: I, § 3, no. 1. Compatible (mapping) with the operation of a monoid: I, § 5, no. 1. Compatible module or multinode structure: II, § 1, no. 14. Complementary minors: III, § 8, no. 6. Component, homogeneous, of an element in a graded group: II, § 11, no. 1. Component of an element in a direct sum: II, § 1, no. 8. Component of an element with respect to a basis: II, § 1, no. 11. Component, S-connected: I, § 7, Exercise 18. Component submodule of a direct sum: II, § 1, no. 6. Composition in M(X): I, § 7, no. 1. Composition, law of: I, § 1, no. 1. Composition of a family with finite support: I, § 2, no. 1. Composition of an ordered sequence: I, § 1, no. 2. Composition of the empty family: I, § 2, no. 1. Composition of words: I, § 7, no. 2. Composition series: I, § 4, no. 7. Condition, maximal (resp. minimal) (set of subgroups satisfying the): I, § 4, Exercise 15. Congruence modulo a rational integer: I, § 4, no. 10. Congruent (elements) modulo an ideal: I, § 8, no. 7. Conjugacy class in a group: I, § 5, no. 4. Conjugate elements in a group: I, § 5, no. 4. Conjugate elements under the operation of a group: I, § 5, no. 4. Conjugate subsets in a group: I, § 5, no. 4. Conjugation, conjugate in a Cayley algebra: III, § 2, no. 4. Conjugation, conjugate in a quadratic algebra: III, § 2, no. 3. Constants of structure of an algebra with respect to a basis: III, § 1, no. 7. Contraction of two indices in a mixed tensor: III, § 5, no. 6. Contragredient of an invertible square matrix: II, § 10, no. 7.

Contragradient of an isomorphism: II, § 2, no. 5. Contravariant tensor: III, § 5, no. 6. Coordinate, barycentric: II, § 9, no. 3. Coordinate form: II, § 2, no. 6. Coordinate of an element with respect to a basis: II, § 1, no. 11. Coordinates, homogeneous (system of), of a point in a projective space: II, § 9, no. 6. Coordinates of a tensor over M with respect to a basis of M: III, § 5, no. 6. Coproduct: III, § 11, no. 1. Coset (right, left) modulo a subgroup: I, § 4, no. 4. Cotranspose of an endomorphism: III, § 8, no. 6. Counital cogebra: III, § 11, no. 2. Counit: III, § 11, no. 2. Covariant tensor: III, § 5, no. 6. Cramer formulae, system: III, § 8, no 7. Cross-ratio: II, § 9, Exercise 11. Crossed homomorphism: I, § 6, Exercise 7. Crossed product: III, § 2, Exercise 11. Cyclic group: I, § 4, no. 10. Cycle of a permutation: I, § 5, no. 7.

Decomposable $p$-vector: III, § 11, no. 13. Decomposition, direct, of a ring: I, § 8, no. 11. Decomposition, reduced decomposition of an element in an amalgamated sum of monoids: I, § 7, no. 3. Defined (group) by generators and relations: I, § 7, no. 6. Defined (monoid) by generators and relations: I, § 7, no. 2. Degree of a homogeneous element in a graded group: II, § 11, no. 1. Degree of a polynomial with respect to the indeterminates X_j such that $j \in J$: III, § 2, no. 9. Degree, total degree, of a monomial: III, § 2, no. 9. Degree, total degree, of an element of a free algebra, of a free associative algebra: III, § 2, no. 7. Degree, total degree, of a polynomial: III, § 2, no. 9. Denominator: I, § 2, no. 4. $\varepsilon$-derivation, inner: III, § 10, no. 6. Derivation, K-derivation: III, § 10, no. 2. (K, $\varepsilon$)-derivation of degree $\delta$, $\varepsilon$-derivation of degree $\delta$: III, § 10, no. 2. $\varepsilon$-derivation of a graded algebra, $\varepsilon$-derivation of a ring: III, § 10, no. 2. Derivation of a ring A into a ring B: III, § 10, no. 2. Derivative, partial: III, § 10, no. 11. Derived (element) from an element of the free algebra by substituting elements for the indeterminates: III, § 2, no. 8.

Derived (element) from an element of the free associative algebra by substituting elements for the indeterminates: III, § 2, no. 8. Derived group of a group: I, § 6, no. 2. Derived (left action, right action) from a law of composition: I, § 3, no. 1. Derived series of a group: I, § 6, no. 4. Determinant, Cauchy: III, § 8, Exercise 5. Determinant of a matrix: III, § 8, no. 3. Determinant of an endomorphism: III, § 8, no. 1. Determinant of a sequence of vectors: III, § 8, no. 1. Determinant, Vandermonde: III, § 8, no. 6. Diagonal elements of a square matrix: II, § 10, no. 7. Diagonal matrix of matrices: II, § 10, no. 7. Diagonal of a square matrix, diagonal matrix: II, § 10, no. 7. Diagram, exact: II, § 1, no. 4. Differences, group of: I, § 2, no. 4. Differences, monoid of: I, § 2, no. 4. K-differential: III, § 10, no. 11. Differential of an element: III, § 10, no. 11. Dihedral group: I, § 6, Exercise 4. Dilatation: II, § 10, Exercise 11. Dimension of a free module: II, § 7, no. 2. Dimension of an affine linear variety: II, § 9, no. 3. Dimension of an affine space: II, § 9, no. 1. Dimension of a projective space: II, § 9, no. 5. Dimension of a vector space: II, § 7, no. 2. Dimorphism: II, § 1, no. 13. Direct decomposition of a ring: I, § 8, no. 11. Direct factor: I, § 4, no. 9. Direct limit: see Limit, direct. Direct product: I, § 4, no. 9. Direct product, internal: I, § 4, no. 9. Direct sum: I, § 4, no. 9. Direct system: see System direct. Direction of an affine linear variety: II, § 9, no. 3. Direction parameters of an affine line: II, § 9, no. 3. Direction vector of an affine line: II, § 9, no. 3. Discriminant ideal of an algebra: III, § 9, no. 5. Discriminant of a finite sequence in an algebra: III, § 9, no. 5. Distributive action: I, § 3, no. 4. Distributive, left distributive, right distributive, law: I, § 3, no. 4. Distributive (mapping) with respect to an index: I, § 3, no. 4. Distributivity of one law of composition with respect to another: I, § 3, no. 5. Divisor, left, right: I, § 8, no. 1.

Divisor of zero, left, right: I, § 8, no. 1. Domain, integral, domain of integrity: I, § 9, no. 2. Double coset with respect to two subgroups: I, § 5, no. 4. Dual bases: II, § 2, nos. 6 and 7. Dual, graded, of a graded module: II, § 11, no. 6. Dual numbers, algebra of: III, § 2, no. 3. Dual of a module: II, § 2, no. 3.

Element, central: I, § 1, no. 5. Element centralizing a subset: I, § 5, no. 3. Element derived from an element of the free algebra by substituting elements for indeterminates: III, § 2, no. 8. Element derived from an element of the free associative algebra by substituting elements for indeterminates: III, § 2, no. 8. Element, free, of a module: II, § 1, no. 11. Element, homogeneous (homogeneous of degree $n$), of a graded group: II, § 11, no. 1. Element, identity: I, § 2, no. 1. Element invariant under an operator: I, § 3, no. 2. Element, isobaric, of a graded group: II, § 11, no. 1. Element, left cancellable, right cancellable, cancellable: I, § 2, no. 2. Element, left invertible, right invertible, invertible: I, § 2, no. 3. Element, left regular, right regular, regular: I, § 2, no. 2. Element normalizing a subset: i, § 5, no. 3. Element, $p$-regular: I, § 6, Exercise 28. Element, $p$-unipotent: I, § 6, Exercise 28. Element, primitive, in a free group: I, § 7, Exercise 26. Element, primitive, of a graded bigebra: III, § 11, no. 8. Element resulting from substituting elements for indeterminates in a free group: I, § 7, no. 5. Element, s-neighbouring: I, § 7, Exercise 18. Element, torsion, of a module: II, § 7, no. 10. Element, unit: I, § 2, no. 1. Element, unit, of an algebra: III, § 1, no. 1. Element, zero: I, § 2, no. 1. Elements congruent modulo an ideal: I, § 8, no. 7. Elements, conjugate, in a group: I, § 5, no. 4. Elements, conjugate, under the operation of a group: I, § 5, no. 4. Elements, diagonal, of a square matrix: II, § 10, no. 7. Elements, linearly dependent (linearly independent) in a module: II, § 1, no. 11. Elements, orthogonal: II, § 2, no. 4. Elements, permutable, elements which commute: I, § 1, no. 5.

Empty matrix: II, § 10, no. 1. Endomorphism: I, § 1, no. 1. Endomorphism of a module: II, § 1, no. 2. Endomorphism of a ring: I, § 8, no. 4. Endomorphism, unimodular: III, § 8, no. 1. Endomorphisms, equivalent, similar: III, § 8, Exercise 26. Ends, number of ends: I, § 7, Exercise 37. Envelope, injective, of a module: II, § 2, no. 1. Equation, linear, homogeneous linear equation, homogeneous linear equation associated with a linear equation: II, § 2, no. 8. Equation of a hyperplane: II, § 7, no. 5. Equation, scalar linear: II, § 2, no. 8. Equations, linear (system of): II, § 2, no. 8. Equations (system of) of a vector subspace: II, § 7, no. 5. Equivalent composition series: I, § 4, no. 7. Equivalent endomorphisms: III, § 8, Exercise 26. Equivalent matrices: II, § 10, no. 9. Even permutation: I, § 5, no. 7. Exact diagram: II, § 1, no. 4. Exact sequence: II, § 1, no. 4. Expansion by a column: III, § 8, no. 6. Expansion by a row: III, § 8, no. 6. Expansion, Laplace: III, § 8, no. 6. Extension, Cayley, of an algebra: III, § 2, no. 5. Extension, central: I, § 6, no. 1. Extension, essential, of a module: II, § 2, Exercise 15. Extension of laws of operation: I, § 5, no. 1. Extension of one group by another: I, § 6, no. 1. Extension of one module by another: II, § 1, no. 4. Extension of scalars (module obtained by): II, § 5, no. 1. Extension of scalars (algebra obtained by): III, § 1, no. 5. Extension, trivial: I, § 6, no. 1. Extension, trivial, of a module: II, § 1, no. 9. Exterior algebra of a module: III, § 7, no. 1. Exterior power, $p$-th, of an endomorphism: III, § 7, no. 4. Exterior power, $p$-th, of a matrix: III, § 8, no. 5. Exterior power, $p$-th, of a module: III, § 7, no. 4. Exterior product of a $p$-vector and a $q$-vector: III, § 7, no. 1. External law of composition: I, § 3, no. 1.

Factor, direct, of a group: I, § 4, no. 9. Factor, direct, of a module: II, § 1, no. 9. Factor of a product: I, § 1, no. 2.

Factors, system of: III, § 2, Exercise 11. Faithfully (monoid operating): I, § 5, no. 1. Faithful module: III, § 1, no. 12. Family, affinely free, affinely related, of points in an affine space: II, § 9, no. 3. Family, basic, free, generating, in a group: I, § 7, no. 6. Family, free, related, of elements of a module: II, § 1, no. 11. Family, generating, of an algebra: III, § 1, no. 2. Family, orthogonal, of projectors: II, § 1, no. 8. Family, projectively free, projectively related, of points in a projective space: II, § 9, no. 7. Fibre product: I, § 4, no. 8. Field: I, § 9, no. 1. Field, commutative, skew field: I, § 9, no. 1. Field of fractions of an integral domain: I, § 9, no. 2. Field of left fractions: I, § 9, Exercise 15. Field of rational numbers: I, § 9, no. 4. Field, projective: II, § 9, no. 9. Finer composition series: I, § 4, no. 7. Finite group: I, § 4, no. 1. Finitely generated group: I, § 7, Exercise 16. Finitely generated module: II, § 1, no. 7. Fixer of a subset of a set: I, § 5, no. 2. Fixing a subset of a set (operator, set of operators): I, § 5, no. 2. Form, canonical bilinear: II, § 2, no. 3. Form, coordinate: II, § 2, no. 6. Form, linear: II, § 2, no. 3. Form, $n$-linear: II, § 3, no. 9. $n$-form: III, § 11, no. 7. Formula, binomial: I, § 8, no. 2. Formula, Leibniz: III, § 10, no. 4. Formulae, Cramer’s: III, § 8, no. 7. Formulae of change of coordinates: II, § 10, no. 8. Formulae, transitivity, of norms and traces: III, § 9, no. 4. Fractions (field of) of an integral domain: I, § 9, no. 2. Fractions, group of, of a monoid: I, § 2, no. 4. Fractions, monoid of, with denominators in S: I, § 2, no. 4. Fractions, ring of, with denominators in S: I, § 8, no. 12. Fractions, total ring of: I, § 8, no. 12. Free algebra, associative algebra, associative and commutative algebra: III, § 2, no. 7. Free algebra of a module: III, § 2, Exercise 13. Free commutative group: I, § 7, no. 5.

Free commutative monoid: I, § 7, no. 7. Free element, family, module, subset, system: II, § 1, no. 11, and (by an abuse of language) II, § 9, no. 7. Free family in a group: I, § 7, no. 6. Free group: I, § 7, no. 5. Free magma: I, § 7, no. 1. Free monoid: I, § 7, no. 2. Free product of algebras: III, § 5, Exercise 6. Free product of groups: I, § 7, no. 3. Free vector in an affine space: II, § 9, no. 1. Freely, group operating: I, § 5, no. 4. Function, linearly affine, affine function: II, § 9, no. 4.

Generated by a family of ordered pairs (equivalent relation): I, § 1, no. 6. Generated by a subset (ideal): I, § 8, no. 6, and III, § 1, no. 2. Generated by a subset (stable subgroup): I, § 4, no. 3. Generated by a subset (stable subset): I, § 1, no. 4. Generated by a subset (subalgebra): III, § 1, no. 2. Generated by a subset (subfield): I, § 9, no. 1. Generated by a subset (submagma): I, § 1, no. 4. Generated by a subset (subring): I, § 8, no. 5. Generated by a subset (unital submagma, submonoid): I, § 2, no. 1. Generating family of a group: I, § 7, no. 6. Generating family of an algebra: III, § 1, no. 2. Generating set, system, of a field: I, § 9, no. 1. Generating set, system, of a magma: I, § 1, no. 4. Generating set, system, of a module: II, § 1, no. 7. Generating set, system, of an ideal: I, § 8, no. 6. Generating set, system, of a ring: I, § 8, no. 5. Generating set, system, of a stable subgroup: I, § 4, no. 3. Generating set, system, of a stable subset: I, § 1, no. 4. Generating set, system, of a unital submagma, of a submonoid: I, § 2, no. 1. Generators of a presentation: I, § 7, no. 6. Graded algebra over a graded ring: III, § 3, no. 1. Graded bigebra: III, § 11, no. 4. Graded bigebra, skew: III, § 11, no. 4. Graded cogebrba: III, § 11, no. 1. Graded group, module, ring: II, § 11, nos. 1 and 2. Graded homomorphism: II, § 11, no. 2. Graded subalgebra: III, § 3, no. 2. Graded subring, submodule, ideal: II, § 11, no. 3. Graded tensor product of type $\Delta_0$: III, § 4, no. 8.

Graduation compatible with a coproduct: III, § 11, no. 1. Graduation compatible with an algebra structure: III, § 3, no. 1. Graduation induced, quotient graduation: II, § 11, no. 3. Graduation of type Δ: II, § 11, no. 1. Graduation, partial, total graduation: II, § 11, nos. 1 and 2. Graduation, trivial: II, § 11, no. 1. Grassmannian: III, § 11, no. 13. Grassmann relations: III, § 11, no. 13. Greatest common divisor (g.c.d.) of two integers: I, § 8, no. 11. Group: I, § 2, no. 3. Group, additive, of a ring: I, § 8, no. 1. Group, affine: I, § 9, no. 4. Group, alternating: I, § 5, no. 7. Group, bicyclic: I, § 6, Exercise 26. Group, bigraded: II, § 11, no. 2. Group, coincidence, of two homomorphisms: I, § 4, no. 8. Group commutator: I, § 6, no. 2. Group, cyclic: I, § 4, no. 10. Group defined by generators and relations: I, § 7, no. 6. Group, derived: I, § 6, no. 2. Group, dihedral: I, § 6, Exercise 4. Group, finite, infinite group: I, § 4, no. 1. Group, finitely generated: I, § 7, Exercise 16. Group, finitely presented: I, § 7, Exercise 16. Group, free commutative, over a set: I, § 7, no. 7 and II, § 1, no. 11. Group, free, over a set: I, § 7, no. 5. Group, graded: II, § 11, no. 1. Group, linear: II, § 2, no. 6. Group, minimal simple: I, § 6, Exercise 27. Group, monogenous: I, § 4, no. 10. Group, multiplicative, of a ring: I, § 8, no. 1. Group, nilpotent, nilpotent group of class $n$: I, § 6, no. 3. Group of differences, group of fractions: I, § 2, no. 4. Group of exponential type: I, § 7, Exercise 39. Group operating faithfully: I, § 5, no. 1. Group operating freely: I, § 5, no. 4. Group operating simply transitively: I, § 5, no. 6. Group operating transitively: I, § 5, no. 5. $p$-group: I, § 6, no. 5. Group, projective: II, § 9, no. 10. Group, residually finite: I, § 5, Exercise 5. Group, solvable, solvable group of class $n$: I, § 6, no. 4. Group, special linear: III, § 8, no. 9.

Group, supersolvable: I, § 6, Exercise 26. Group, symmetric: I, § 4, no. 1. Group, unimodular: III, § 8, no. 9. Group with operators, Abelian, commutative: I, § 4, no. 2. Group with operators: I, § 4, no. 2. Group with operators, product: I, § 4, no. 8. Group with operators, product (or internal product), of a family of quotient groups: I, § 4, no. 8. Group with operators, quotient: I, § 4, no. 4. Group with operators, simple: I, § 4, no. 4. Groupoid: I, § 4, Exercise 23.

Hall’s Theorem: I, § 6, Exercise 10. Hamel basis: II, § 7, no. 1. Hamiltonian quaternions: III, § 2, no. 5. Homogeneous element in a graded group: II, § 11, no. 1. Homogeneous G-set: I, § 5, no. 5. Homogeneous linear equation, linear system: II, § 2, no. 8. Homogeneous subset of degree $p$ in a formal power series: III, § 2, no. 11. Homogeneous subset of degree $p$ with respect to certain indeterminates in a formal power series: III, § 2, no. 11. Homomorphism, algebra: III, § 1, no. 1. Homomorphism, A-module, A-homomorphism: II, § 1, no. 2. Homomorphism, central ring: II, § 5, no. 3. Homomorphism, crossed: I, § 6, Exercise 7. Homomorphism, essential: II, § 2, Exercise 15. Homomorphism for two laws of composition: I, § 1, no. 1. Homomorphism, graded: II, § 11, no. 2. Homomorphism, graded algebra: III, § 3, no. 1. Homomorphism, group: I, § 4, no. 1. Homomorphism, monoid: I, § 2, no. 1. Homomorphism, M-set: I, § 5, no. 1. Homomorphism, multinode: II, § 1, no. 14. Homomorphism of groups with operators: I, § 4, no. 2. $\phi$-homomorphism: I, § 3, no. 1. Homomorphism, projection: I, § 4, no. 8. Homomorphism, ring: I, § 8, no. 4. Homomorphism, trivial: I, § 2, no. 1. Homomorphism, unital: I, § 2, no. 1. Homomorphism, unital algebra: III, § 1, no. 1. Homothety: I, § 4, no. 2 and II, § 1, no. 1. Homothety, central: II, § 1, no. 2 and III, § 9, Exercise 6. Hyperplane, affine: II, § 9, no. 3.

Hyperplane at infinity: II, § 9, no. 8. Hyperplane passing through 0 in a vector space: II, § 7, no. 3. Hyperplane, projective: II, § 9, no. 7. Hyperplane, projective, taken as hyperplane at infinity: II, § 9, no. 10.

Ideal, discriminant: III, § 9, no. 5. Ideal, graded: II, § 11, no. 3 and III, § 3, no. 2. Ideal, irreducible two-sided: I, § 8, Exercise 11. Ideal, left, right ideal, two-sided ideal in an algebra: III, § 1, no. 2. Ideal, left, right ideal, two-sided ideal in a ring: I, § 8, no. 6. Ideal, maximal: I, § 8, no. 6. Ideal, prime: I, § 9, no. 3. Ideal, principal: I, § 8, no. 6. Ideal of relators: III, § 2, no. 8. Ideal, zero: I, § 8, no. 6. Idempotent: I, § 1, no. 4. Identities, Jacobi, between minors of a determinant: III, § 11, Exercise 9. Identities, polynomials: III, § 2, no. 9. Identity element: I, § 2, no. 1. Identity, Jacobi: III, § 10, no. 6. Identity, Redei: III, § 8, Exercise 25. Image (inverse image) of a projective linear variety under a projective mapping: II, § 9, no. 10. Image of a homomorphism: II, § 1, no. 3. Indeterminate, indeterminate of index i: I, § 7, no. 5 and III, § 2, no. 7. Index of a subgroup: I, § 4, no. 4. Induced action: I, § 3, no. 2. Induced graduation: II, § 11, no. 3. Induced K'-structure: II, § 8, no. 1. Induced law: I, § 1, no. 4. Infinity, hyperplane at: II, § 9, no. 8. Infinity, points at: II, § 9, no. 8. Inner automorphism of a group: I, § 5, no. 3. Inner automorphism of a ring: I, § 8, no. 4. Inner product, left, right: III, § 11, nos. 6 and 7. Integer, negative, positive, strictly negative, strictly positive: I, § 2, no. 5. Integer, rational: I, § 2, no. 5. Integers, rational, modulo a (ring of): I, § 8, no. 11. Integers, relatively prime: I, § 8, no. 11. Integral domain: I, § 9, no. 2. Integrity, domain of: I, § 9, no. 2. Internal direct product, restricted sum: I, § 4, no. 9. Invariant element: I, § 5, no. 2.

Invariant subgroup, stable subgroup: I, § 4, no. 4. Inverse, left inverse, right inverse, element: I, § 2, no. 3. Inverse, left, right inverse, of a linear mapping: II, § 1, no. 9. Inverse limit: see Limit, inverse. Inverse system: see System, inverse. Inversion in a bigebra: III, § 11, Exercise 4. Inversion of a permutation: I, § 5, no. 7. Invertible, left invertible, right invertible, element: I, § 2, no. 3. Invertible, left, right invertible, linear mapping: II, § 1, no. 9. Invertible square matrix: II, § 10, no. 7. Irreducible ideal: I, § 8, Exercise 11. Isobaric element: II, § 11, no. 1. Isomorphic magmas: I, § 1, no. 1. Isomorphism, magma: I, § 1, no. 1.

Jacobi identity: III, § 10, no. 6. Jordan-Hölder series: I, § 4, no. 7. Jordan-Hölder Theorem: I, § 4, no. 7. Juxtaposition: I, § 7, no. 2.

Kernel of a group homomorphism: I, § 4, no. 5. Kernel of a linear mapping: II, § 1, no. 3. Kronecker symbol: II, § 1, no. 11. Krull’s Theorem: I, § 8, no. 6.

Laplace expansion: III, § 8, no. 6. Law and equivalence relation, compatible: I, § 1, no. 6. Law, associative: I, § 1, no. 3. Law, commutative: I, § 1, no. 5. Law, group: I, § 4, no. 1. Law, induced: I, § 1, no. 4. Law, left distributive, right distributive, distributive with respect to another I, § 3, nos. 4 and 5. Law not everywhere defined: I, § 1, no. 1. Law of composition: I, § 1, no. 1. Law of left, right, operation of a monoid on a set: I, § 5, no. 1. Law of right, left action associated with an action: I, § 3, no. 1. Law, opposite: I, § 1, no. 1. Law, quotient: I, § 1, no. 6. Law written additively, multiplicatively: I, § 1, no. 1. Least common multiple (l.c.m.) of two integers: I, § 8, no. 11. Leibniz formula: III, § 10, no. 4. Lemma, Zassenhaus’s: I, § 4, no. 7.

Length of a decomposition in an amalgamated sum: I, § 7, no. 3. Length of a group: I, § 4, no. 7. Length of a module: II, § 1, no. 10. Length of an element in a free group: I, § 7, Exercise 19. Length of an element in a free magma: I, § 7, no. 1. Length of a word: I, § 7, no. 2. Limit, direct, of $A_\alpha$-algebras: III, § 1, no. 6. Limit, direct, of $A_\alpha$-modules: II, § 6, no. 2. Limit, direct, of actions: I, § 10, no. 4. Limit, direct, of graded algebras: III, § 3, no. 3. Limit, direct, of groups, of groups with operators, of monoids: I, § 10, no. 3. Limit, direct, of magmas: I, § 10, no. 3. Limit, direct, of rings: I, § 10, no. 3. Limit, inverse, of $A_\alpha$-algebras: III, § 1, no. 6. Limit, inverse, of $A_\alpha$-modules: II, § 6, no. 1. Limit, inverse, of groups, of groups with operators, of monoids: I, § 10, no. 1. Limit, inverse, of magmas: I, § 10, no. 1. Limit, inverse, of rings: I, § 10, no. 1. Line, affine: II, § 9, nos. 1 and 3. Line (passing through 0) in a vector space: II, § 7, no. 3. Line, projective: II, § 9, no. 5. Linear equation, system: II, § 2, no. 8. Linear equation, system, homogeneous: II, § 2, no. 8. Linear form: II, § 2, no. 3. Linear group: II, § 2, no. 6. Linear group, special: III, § 8, no. 9. Linear mapping: II, § 1, no. 2. Linear mapping, function, affine: II, § 9, no. 4. Linear mapping, projective: II, § 9, no. 10. Linear problem: II, § 2, no. 8. Linear variety: II, § 9, nos. 3, 7 and 11. Linear variety, affine: II, § 9, no. 3. Linear variety, projective: II, § 9, nos. 7 and 11. Linearly dependent, linear independent, elements: II, § 1, no. 11. Linearly disjoint subalgebras: III, § 4, no. 4. Linearity, principle of extension by: II, § 1, no. 7.

Magma: I, § 1, no. 1. Magma, associative: I, § 1, no. 3. Magma, commutative: I, § 1, no. 5. Magma, defined by generators and relations: I, § 7, no. 1. Magma, free, over a set: I, § 7, no. 1. Magma of mappings into a magma: I, § 1, no. 1.

Magma, opposite: I, § 1, no. 1. Magma, product: I, § 1, no. 1. Magma, quotient: I, § 1, no. 6. Magma, unital: I, § 2, no. 1. Magmas, isomorphic: I, § 1, no. 1. Mapping, affine linear, affine: II, § 9, no. 4. Mapping, alternating multilinear: III, § 7, no. 4. Mapping, biadditive, $\mathbf{Z}$-bilinear: II, § 3, no. 1. Mapping, C-bilinear: II, § 3, no. 5. Mapping compatible with an action: I, § 3, no. 1. Mapping compatible with the operation of a monoid: I, § 5, no. 1. Mapping distributive with respect to a variable: I, § 3, no. 4. Mapping, linear, A-linear: II, § 1, no. 2. Mapping, linear, $A_s(T) \to E$ determined by a mapping $T \to E$: II, § 1, no. 11. Mapping, linear, associated with an affine mapping: II, § 9, no. 4. Mapping, multiadditive, $\mathbf{Z}$-multilinear: II, § 3, no. 9. Mapping, C-multilinear: II, § 3, no. 9. Mapping, orbital: I, § 5, no. 4. Mapping, projective linear, projective: II, § 9, no. 10. Mapping, right invertible linear, left invertible linear: II, § 1, no. 9. Mapping, semi-linear: II, § 1, no. 13. Mapping, symmetric multilinear: III, § 6, no. 3. Matrices differing only by the order of the columns, rows: II, § 10, no. 9. Matrices, equivalent: II, § 10, no. 9. Matrices, similar square: II, § 10, no. 9. Matrix, contragredient, of an invertible matrix: II, § 10, no. 7. Matrix, diagonal: II, § 10, no. 7. Matrix, empty: II, § 10, no. 1. Matrix, invertible: II, § 10, no. 7. Matrix, lower triangular, upper triangular: II, § 10, no. 7. Matrix, matrix of type $(p, q)$: II, § 10, no. 1. Matrix, monomial: II, § 10, no. 7. Matrix obtained by bordering a matrix: II, § 10, no. 1. Matrix obtained by suppressing columns, rows: II, § 10, no. 1. Matrix of a linear mapping with respect to two bases: II, § 10, no. 4. Matrix of a linear system: II, § 10, no. 4. Matrix of an element with respect to a basis: II, § 10, no. 4. Matrix of an endomorphism with respect to a basis: II, § 10, no. 7. Matrix of a permutation: II, § 10, no. 7. Matrix of a semi-linear mapping with respect to two bases: II, § 10, no. 6. Matrix of passing from one basis to another: II, § 10, no. 8. Matrix, scalar: II, § 10, no. 7.

Matrix, square, square matrix of order $n$: II, § 10, no. 7. Matrix, unimodular: III, § 8, no. 4. Matrix units: II, § 10, no. 3. Matrix with only zeros below (above) the diagonal: II, § 10, no. 7. Matrix, zero: II, § 10, no. 2. Maximal ideal: I, § 8, no. 6. G-mean: I, § 6, Exercise 8. Minimal normal stable subgroups: I, § 4, Exercise 15. Minimal simple group: I, § 6, Exercise 27. Minor, minor of order $p$ of a matrix: III, § 8, no. 5. Minors, complementary: III, § 8, no. 6. Mixed tensor: III, § 5, no. 6. Module, bigraded: II, § 11, no. 2. Module, divisible: II, § 7, Exercise 33. Module, dual: II, § 2, no. 3. Module, faithful: II, § 1, no. 12. Module, faithful, associated with a module: II, § 1, no. 12. Module, free: II, § 1, no. 11. Module, graded free: II, § 11, no. 2. Module, graded, graded module with positive degrees: II, § 11, no. 2. Module, graded quotient: II, § 11, no. 3. Module, indecomposable: II, § 2, Exercise 21. Module, injective: II, § 2, Exercise 11. Module, left, right module, A-module: II, § 1, no. 1. Module, monogeneus: II, § 1, no. 12. Module of finite length: II, § 1, no. 10. Module of formal linear combinations: II, § 1, no. 11. Module of linear relations: II, § 1, no. 11. Module over an algebra: III, § 4, no. 3. Module, product: II, § 1, no. 5. Module, projective: II, § 2, no. 2. Module, quotient: II, § 1, no. 3. Module, reflexive: II, § 2, no. 7. Module, torsion-free, over an integral domain: II, § 7, no. 10. Module, torsion, over an integral domain: II, § 7, no. 10. Monogeneus group: I, § 4, no. 10. Monogenous module: II, § 1, no. 2. Monoid: I, § 2, no. 1. Monoid defined by generators and relations: I, § 7, no. 2. Monoid, free, over a set: I, § 7, no. 2. Monoid of differences: I, § 2, no. 4. Monoid of fractions with denominators in S: I, § 2, no. 4. Monoid operating faithfully on a set: I, § 5, no. 1.

Monoidal sum: I, § 7, no. 3. Monomial: III, § 2, no. 9. Morphism, algebra: III, § 1, no. 1. Morphism, A-module: II, § 1, no. 2. Morphism, cogebra: III, § 11, no. 1. Morphism, graded bigebra, left graded bigebra morphism: III, § 11, no. 4. Morphism, graded cogebra: III, § 11, no. 1. Morphism, magma: I, § 1, no. 1. Morphism, monoid: I, § 2, no. 1. Morphism of extensions: I, § 6, no. 1. Morphism, ring: I, § 8, no. 4. Morphism, unital algebra: III, § 4, no. 5. Morphism, unital magma: I, § 2, no. 1. ϕ-morphism (ϕ a homomorphism of one monoid of operators into another): I, § 5, no. 1. ϕ-morphism (ϕ a mapping of one set of operators into another): I, § 3, no. 1. M-morphism (M a monoid of operators): I, § 5, no. 1. Ω-morphism (Ω a set of operators): I, § 3, no. 1. Multiadditive, Z-multilinear, mapping: II, § 3, no. 9. Multidegree in the free algebra of a monoid: III, § 2, Exercise 13. Multiindex: I, § 7, no. 7. Multilinear form: II, § 3, no. 9. C-multilinear mapping: II, § 3, no. 9. Multimodule: II, § 1, no. 14. Multimodule, quotient: II, § 1, no. 14. Multiple, left, right: I, § 8, no. 1. Multiplication: I, § 1, no. 1. Multiplication in an algebra. III, § 1, no. 1. Multiplication table: III, § 1, no. 7. Multiplicative group of a ring: I, § 8, no. 1.

Negative of an element: I, § 2, no. 3. Negative rational integer: I, § 2, no. 5. Negative rational number: I, § 9, no. 4. S-neighbouring element: I, § 7, Exercise 18. Nielson-Schreier Theorem: I, § 7, Exercise 20. Nilpotent group, nilpotent group of class n: I, § 6, no. 3. Norm, Cayley: III, § 2, no. 4. Norm in a quadratic algebra: III, § 2, no. 3. Norm of an element in a K-algebra relative to K: III, § 9, no. 3. Norm of a scalar relative to a module: III, § 9, no. 1. Normal stable subgroup: I, § 4, no. 4.

Normal subgroup: I, § 4, no. 4. Normalizer: I, § 5, no. 3. Normalizing (element, subset) a subset: I, § 5, no. 3. Null element: I, § 2, no. 1. Number, prime: I, § 4, no. 10. Number, rational: I, § 9, no. 4. Number (rational), negative, positive, strictly negative, strictly positive: I, § 9, no. 4. Numerator: I, § 2, no. 4.

Octonions, Cayley: III, Appendix, no. 3. Octonions of type ($\alpha, \beta, \gamma, \delta$) (algebra of): III, Appendix, no. 3. Odd permutation: I, § 5, no. 7. Operation by left, right, translation: I, § 5, no. 1. Operation, left, right (laws of): I, § 5, no. 1. Operation, left, right, of a monoid: I, § 5, no. 1. Operation, simply transitive: I, § 5, no. 6. Operation, transitive: I, § 5, no. 5. Operation, trivial: I, § 5, no. 2. Operator: I, § 3, no. 1. Opposite algebra: III, § 1, no. 1. Opposite cogebra: III, § 11, no. 1. Opposite law: I, § 1, no. 1. Opposite magma: I, § 1, no. 1. Opposite to an M-set, $M^0$-set: I, § 5, no. 1. Opposite ring: I, § 8, no. 3. Orbit: I, § 5, no. 4. Orbital mapping: I, § 5, no. 4. Order, element of infinite: I, § 4, no. 10. Order of a cycle: I, § 5, no. 7. Order of a formal power series with respect to certain indeterminates: III, § 2, no. 11. Order of a group: I, § 4, no. 1. Order of an element in a group: I, § 4, no. 10. Order of a square matrix: II, § 10, no. 7. Order, total order, of a formal power series: III, § 2, no. 11. Ordered sequence: I, § 1, no. 2. Ordered sequences, similar: I, § 1, no. 2. Origin: I, § 2, no. 1. Origin, choice of, in an affine space: II, § 9, no. 1. Orthogonal elements, sets: II, § 2, no. 3. Orthogonal family of projectors: II, § 1, no. 8. Orthogonal, submodule, to a subset of E (resp. $E^*$): II, § 2, no. 4.

Parallel linear varieties: II, § 9, no. 3. Parallelogram: II, § 9, Exercise 1. Parameters, direction, of an affine line: II, § 9, no. 3. Partial derivative: III, § 10, no. 11. Partial graduation: II, § 11, no. 1. Passage, matrix of: II, § 10, no. 8. Permutable elements: I, § 1, no. 5. Permutation, even, odd: I, § 5, no. 7. Plane, affine: II, § 9, nos. 1 and 3. Plane passing through 0 in a vector space: II, § 7, no. 3. Plane, projective: II, § 9, no. 5. Point of an affine space: II, § 9, no. 1. Point of a projective space: II, § 9, no. 5. Points, affinely independent: II, § 9, no. 3. Points at infinity: II, § 9, no. 8. Polynomial, characteristic, of an element in a K-algebra: III, § 9, no. 3. Polynomial, characteristic, of an endomorphism: III, § 8, no. 11. Polynomial, characteristic, of a scalar with respect to a module: III, § 9, no. 1. Polynomial containing no term in X^v: III, § 2, no. 9. Polynomial identities: III, § 2, no. 9. Polynomial of degree n: III, § 2, no. 9. Polynomial relators: III, § 2, no. 9. Polynomial with no constant term: III, § 2, no. 9. Polynomial with respect to a family of indeterminates, with coefficients in a ring: III, § 2, no. 9. Positive rational integer: I, § 2, no. 5. Positive rational number: I, § 9, no. 4. Power, exterior, of a linear mapping: III, § 7, no. 4. Power, exterior, of a matrix: III, § 8, no. 5. Power, exterior, of a module: III, § 7, no. 4. Power, n-th, under an associative law: I, § 1, no. 3. Power, symmetric, of a linear mapping, of a module: III, § 6, no. 3. Power, tensorial, of a linear mapping: III, § 5, no. 2. Power, tensorial, of a module: III, § 5, no. 1. Presentation of a group: I, § 7, no. 6. Presentation of an algebra: III, § 2, no. 8. Presented, finitely (group): I, § 7, Exercise 16. Preservation when passing to the quotient: I, § 1, no. 6. Prime ideal: I, § 9, no. 3. Prime number: I, § 4, no. 10. Prime, relatively (integers): I, § 8, no. 10. Primitive element in a free group: I, § 7, Exercise 26.

Primitive element in a free magma generated by a single element: I, § 7, Exercise 7. Primitive element of a graded bigebra: III, § 11, no. 8. Principal G-set, homogeneous: I, § 5, no. 6. Principal ideal: I, § 8, no. 6. Principal series: I, § 4, Exercise 17. Principal set under G, homogeneous: I, § 5, no. 6. Principle of extension by linearity: II, § 1, no. 7. Problem, linear: II, § 2, no. 8. Product algebra: III, § 1, no. 4. Product, block, of matrices: II, § 10, no. 5. Product, crossed: III, § 2, Exercise 11. Product, exterior, of a $p$-vector and a $q$-vector: III, § 7, no. 1. Product, external semi-direct, of G by F relative to $\tau$: I, § 6, no. 1. Product, fibre, of groups with operators: I, § 4, no. 8. Product, fibre, of modules: II, § 1, Exercise 4. Product, free, of algebras: III, § 5, Exercise 6. Product, free, of groups: I, § 7, no. 5. Product, graded tensor, of two graded modules: II, § 11, no. 5. Product, graded tensor, of graded algebras: III, § 4, nos. 7 and 9. Product group, internal product group, of a family of quotient groups: I, § 4, no. 8. Product group with operators: I, § 4, no. 8. Product, internal direct, direct product, product, of subgroups: I, § 4, no. 9. Product magma: I, § 1, no. 1. Product of K'-structures: II, § 8, no. 3. Product of laws of composition: I, § 1, no. 1. Product of matrices calculated according to a mapping: II, § 10, no. 2. Product of matrices calculated in a ring: II, § 10, no. 3. Product of modules: II, § 1, no. 5. Product of multimodules: II, § 1, no. 14. Product of an operator and an element: I, § 3, no. 1. Product of an ordered sequence: I, § 1, no. 2. Product of two elements: I, § 1, no. 1. Product of two-sided ideals: I, § 8, no. 9. Product, (right, left) inner: III, § 11, nos. 6 and 7. Product ring: I, § 8, no. 10. Product, symmetric, of multilinear mappings: III, § 11, no. 2. Product, tensor, of a family of $\mathbf{Z}$-modules relative to a triple $(c, p, q)$: II, § 3, no. 9. Product, tensor, of algebras: III, § 4, no. 1. Product, tensor, of an infinite family of algebras: III, § 4, no. 5. Product, tensor, of bases of algebras: III, § 4, no. 5.

Product, tensor, of cogebras: III, § 11, no. 1. Product, tensor, of two bases: II, § 3, no. 7. Product, tensor, of two elements: II, § 3, no. 1. Product, tensor, of two linear mappings: II, § 3, no. 2. Product, tensor, of two matrices over a commutative ring: II, § 10, no. 10 Product, tensor, of two modules: II, § 3, no. 1. Product, tensor, of two multimodules: II, § 3, no. 4. Product, tensor, of two semi-linear mappings: II, § 3, no. 3. Projection homomorphism: I, § 4, no. 8. Projective field: II, § 9, no. 9. Projective group: II, § 9, no. 10. Projective hyperplane, plane: II, § 9, no. 7. Projective line: II, § 9, no. 5. Projective mapping: II, § 9, no. 10. Projective module: II, § 2, no. 2. Projective space: II, § 9, nos. 5 and 11. Projectively free, projectively related, family: II, § 9, no. 7. Projector: II, § 1, no. 8. Pseudomodule, left, right: II, Appendix, no. 2. Pseudo-ring: I, § 8, no. 1. Pseudo-ring with zero square: I, § 8, no. 3. Pure $p$-vector: III, § 11, no. 13. Pure quaternion: III, § 2, Exercise 3.

Quadratic algebra: III, § 2, no. 3. Quasi-group: I, § 3, Exercise 6. Quaternion, pure: III, § 2, Exercise 3. Quaternion algebra: III, § 2, no. 5. Quaternionic group: I, § 6, Exercise 4. Quotient algebra: III, § 1, no. 2. Quotient graduation: II, § 11, no. 3. Quotient group with operators: I, § 4, no. 4. Quotient law: I, § 1, no. 6. Quotient magma: I, § 1, no. 11. Quotient module, vector space: II, § 1, no. 3. Quotient multinode: II, § 1, no. 14. Quotient of an action: I, § 3, no. 3. Quotient ring: I, § 8, no. 7. Quotients of a composition series of a group with operators: I, § 4, no. 7.

Rank of a free group: I, § 7, Exercise 14. Rank of a linear mapping of vector spaces: II, § 7, no. 4. Rank of a linear system over a field: II, § 7, no. 6.

Rank of a matrix over a field: II, § 10, no. 12. Rank of an affine linear mapping: II, § 9, no. 4. Rank of an element in a tensor product of vector spaces: II, § 7, no. 8. Rank of a semi-linear mapping of vector spaces: II, § 7, no. 4. Rank of a subset of a module over an integral domain: II, § 7, no. 10. Rank of a subset of a vector space: II, § 7, no. 2. Ratio of a homothety: II, § 1, no. 1. Rational integer: I, § 2, no. 5. Rational number: I, § 9, no. 4. Rational numbers, field of : I, § 9, no. 4. Rational over a subfield (linear form): II, § 8, no. 4. Rational over a subfield (linear mapping): II, § 8, no. 3. Rational over a subfield (subspace, vector): II, § 8, no. 2. Reduced decomposition of an element of an amalgamated sum: I, § 7, no. 3. Reflexive module: II, § 2, no. 7. Regular, left regular, right regular, element: I, § 2, no. 2. p-regular element: I, § 6, Exercise 28. Related system, subset: II, § 1, no. 11, and § 9, no. 7. Relation, equivalence, compatible with a law of composition: I, § 1, no. 6. Relation, equivalence, compatible with an action: I, § 3, no. 3. Relation, equivalence, generated by a family of ordered pairs: I, § 1, no. 6. Relation, equivalence, left, right, compatible with a law of composition: I, § 3, no. 3. Relations, commutativity, in a multiplication table: III, § 1, no. 7. Relations, Grassmann: III, § 11, no. 13. Relations, module of linear: II, § 1, no. 11. Relatively prime integers: I, § 8, no. 11. Relator: I, § 7, no. 6 and III, § 2, no. 8. Relator, universal: III, § 2, no. 8. Relators, ideal of: III, § 3, no. 8. Relators of a presentation: I, § 7, no. 6 and III, § 2, no. 8. Relators, polynomial: III, § 2, no. 9. Residue of a semi-group: I, § 2, Exercise 11. Residually finite group: I, § 5, Exdrcise 5. Restricted algebra of a monoid: III, § 2, no. 10. Restricted sum, internal: I, § 4, no. 9. Restricted sum of groups: I, § 4, no. 9. Restricted sum of groups with respect to subgroups: I, § 4, no. 9. Restriction of scalars (algebra obtained by): III, § 1, no. 5. Resulting (element) from substituting elements for indeterminates: I, § 7, no. 5. Retraction of an extension: I, § 6, no. 1.

Right hand side of a linear equation, right hand sides of a linear system: II, § 2, no. 8. Ring: I, § 8, no. 1. Ring, Boolean: I, § 9, Exercise 8. Ring, commutative: I, § 8, no. 1. Ring (graded, graded with positive degrees, bigraded): II, § 11, no. 2. Ring, graded quotient: II, § 11, no. 3. Ring obtained by the adjunction of a unit element: II, Appendix, no. 1. Ring of endomorphisms of a commutative group: I, § 8, no. 4. Ring of fractions with denominators in S: I, § 8, no. 12. Ring of integers modulo n: I, § 8, no. 11. Ring of polynomials which are not commutative relative to an endomorphism and a derivation: III, § 10, Exercise 3. Ring, opposite: I, § 8, no. 3. Ring, product: I, § 8, no. 10. Ring, quotient: I, § 8, no. 7. Ring, total, of fractions: I, § 8, no. 2. Ring, zero: I, § 8, no. 3. Row of a matrix: II, § 10, no. 1. Rule, sign: I, § 8, no. 1.

Scalar: II, § 1, no. 1. Scalar linear equation: II, § 2, no. 8. Scalar matrix: II, § 10, no. 7. Schreier’s Theorem on composition series: I, § 4, no. 7. Section of an extension: I, § 6, no. 2. Semi-direct product of groups: I, § 6, no. 1. Semi-group, left, right: I, § 2, Exercise 11. Semi-homomorphism, algebra, algebra ρ-homomorphism: III, § 1, no. 5. Semi-linear mapping: II, § 1, no. 13. Sequence, exact: II, § 1, no. 4. Sequence, ordered: I, § 1, no. 2. Sequence, split exact: II, § 1, no. 9. Sequences, similar ordered: I, § 1, no. 2. Series, algebra of formal power: III, § 2, no. 11. Series, composition: I, § 4, no. 6. Series, derived: I, § 6, no. 4. Series, equivalent decomposition: I, § 4, no. 7. Series, finer composition: I, § 4, no. 7. Series, formal power: III, § 2, no. 11. Series, Jordan-Hölder: I, § 4, no. 7. Series, lower central: I, § 6, no. 3. Series, normal: I, § 4, Exercise 17.

Series, principal: I, § 4, Exercise 17. G-set, homogeneous (G a group of operators): I, § 5, no. 5. G-set, homogeneous principal, homogeneous principal set under G: I, § 5, no. 6. Set, generating, of a group: I, § 4, no. 3. Set, generating, of a magma: I, § 1, no. 4. M-set (M a monoid of operators): I, § 5, no. 1. M^0-set opposite to an M-set: I, § 5, no. 1. Set of degrees of a graded group: II, § 11, no. 1. Set of subgroups satisfying the maximal condition, the minimal condition: I, § 4, Exercise 15. Sets, orthogonal: II, § 2, no. 4. G-sets, weakly equivalent: I, § 5, Exercise 26. Shift, shifting: II, § 11, no. 2. Signature of a permutation: I, § 5, no. 7. Sign of a rational number: I, § 9, no. 4. Similar endomorphisms: III, § 8, Exercise 26. Similar matrices: II, § 10, no. 9. Similar ordered sequences: I, § 1, no. 2. Simple group: I, § 4, no. 4. Simply transitive operation: I, § 5, no. 6. Skew field: I, § 9, no. 1. Skew graded bigebra: III, § 11, no. 4. Skew-symmetric tensor: III, § 7, no. 4. Skew-symmetrized tensor: III, § 7, no. 4. Skew tensor product of graded algebras: III, § 4, nos. 7 and 8. Solution of a linear equation, of a linear system: II, § 2, no. 8. Solution, trivial, zero solution of a homogeneous linear equation: II, § 2, no. 8. Solvable group, solvable group of class n: I, § 6, no. 4. Space, affine, attached to a vector space: II, § 9, no. 1. Space, canonical projective, associated with a vector space: II, § 9, no. 8. Space, projective: II, § 9, nos. 5 and 11. Space, projective, derived from a vector space: II, § 9, no. 5. Space, quotient vector, quotient space: II, § 1, no. 3. Space, right, left, vector, over a field: II, § 1, no. 1. Space, vector, associated with a module over an integral domain: II, § 7, no. 10. Space, vector, obtained by taking an origin in an affine space: II, § 9, no. 1. Space, vector, of translations of an affine space: II, § 9, no. 1. Split exact sequence: II, § 1, no. 10. Stabilizing (operator, set of operators) a subset: I, § 5, no. 2. Stabilizing, strictly (operator, set of operators), a subset: I, § 5, no. 2. Stable subgroup of a group with operators: I, § 4, no. 3.

Stable subset: I, § 1, no. 4. Strictly negative, strictly positive, rational integer: I, § 2, no. 5. Strictly negative, strictly positive, rational number: I, § 9, no. 4. Strict stabilizer: I, § 5, no. 2. Strictly stabilizing a subset: I, § 5, no. 2. Strict transporter: I, § 5, no. 2. Strongly associative subset: III, Appendix, no. 1. Structures, module (multimodule), compatible: II, § 1, no. 14. Structure, projective space: II, § 9, no. 11. K'-structure, induced: II, § 8, no. 2. K'-structure on a vector K-space: II, § 8, no. 1. K'-structure, product: II, § 8, no. 3. Subalgebra: III, § 1, no. 2. Subalgebra generated by a subset: III, § 1, no. 2. Subalgebra, graded: III, § 3, no. 2. Subfield: I, § 9, no. 1. Subfield generated by a subset: I, § 9, no. 1. Subgroup, characteristic: I, § 5, no. 3. Subgroup, invariant stable, invariant subgroup: I, § 4, no. 4. Subgroup, normal stable, normal subgroup: I, § 4, no. 4. Subgroup, stable, generated by a subset: I, § 4, no. 3. Subgroup, stable subgroup: I, § 4, no. 3. Subgroup, Sylow, Sylow $p$-subgroup: I, § 6, no. 6. Submagma: I, § 1, no. 4. Submagna generated by a subset: I, § 1, no. 4. Submagma, unital: I, § 2, no. 1. Submagma, unital, generated by a subset: I, § 2, no. 1. Submatrix: II, § 10, no. 1. Submodule: II, § 1, no. 3. Submodule, component, of a direct sum of modules: II, § 1, no. 6. Submodule generated by a family: II, § 1, no. 7. Submodule, graded: II, § 11, no. 3. Submodule, irreducible: II, § 2, Exercise 16. Submodule, orthogonal (or totally orthogonal) to a subset of E (resp. E*) II, § 2, no. 4. Submodule, torsion, of a module over an integral domain: II, § 7, no. 10. Submodule, zero: II, § 1, no. 3. Submodules, supplementary: II, § 1, no. 9. Submultimodule: II, § 1, nos. 14. Subring: I, § 8, no. 5. Subring generated by a subset: I, § 8, no. 5. Subring, graded: II, § 11, no. 3. Subset, affine: II, § 9, no. 3.

Subset centralizing a subset: I, § 5, no. 3. Subset, free, related subset: II, § 1, no. 11. Subset, homogeneous, of degree $p$ with respect to certain indeterminates in a formal power series: III, § 2, no. 11. Subset normalizing a subset: I, § 5, no. 3. Subset, stable: I, § 1, no. 4 and § 3, no. 2. Subset, stable, generated by a subset: I, § 1, no. 4 and § 3, no. 2. Subset, strongly associative: III, Appendix, no. 1. Subset, symmetric: I, § 4, no. 1. Subsets, conjugate: I, § 5, no. 4. Subspaces associated with a homogeneous element of the exterior algebra: III, § 7, no. 2. Subspace associated with a homogeneous element of the symmetric algebra: III, § 6, no. 2. Subspace associated with a homogeneous element of the tensor algebra: III, § 5, no. 2. Subspaces associated with an element of a tensor product of vector spaces: II, § 7, no. 8. Subspace rational over a subfield: II, § 8, no. 2. Subspace, vector, subspace: II, § 1, no. 3. Sum, amalgamated, of modules: II, § 1, Exercise 5. Sum, amalgamated, of monoids: II, § 7, no. 3. Sum, direct: I, § 4, no. 9. Sum, direct, of a family of submodules: II, § 1, no. 8. Sum, external direct, of a family of submodules: II, § 1, no. 6. Sum, internal restricted, of subgroups: I, § 4, no. 9. Sum, monoidal: I, § 7, no. 3. Sum of a family of elements of finite support: I, § 2, no. 1. Sum of a family of left ideals, of right ideals: I, § 8, no. 6. Sum of a family of submodules: II, § 1, no. 7. Sum of an ordered sequence: I, § 1, no. 2. Sum of two elements: I, § 1, no. 1. Sum of two matrices: II, § 10, no. 2. Sum, restricted, of groups with respect to subgroups, restricted sum of groups: I, § 4, no. 9. Supersolvable group: I, § 6, Exercise 26. Supplementary submodules: II, § 1, no. 9. Support of a cycle: I, § 5, no. 7. Support of a family: I, § 2, no. 1. Suppress columns, rows, in a matrix: II, § 10, no. 1. Sylow subgroup: I, § 6, no. 6. Symbol, Kronecker: II, § 1, no. 11. Symmetric algebra of a module: III, § 6, no. 1.

Symmetric group: I, § 4, no. 1. Symmetric multilinear mapping: III, § 6, no. 3. Symmetric power of a linear mapping: III, § 6, no. 3. Symmetric power of a module: III, § 6, no. 3. Symmetric product of two multilinear mapping: III, § 11, no. 2. Symmetric subset in a group: I, § 4, no. 1. Symmetric tensor: III, § 6, no. 3. Symmetrization of a tensor: III, § 6, no. 3. System, direct, of graded algebras: III, § 3, no. 3. System, direct, of magmas: I, § 10, no. 3. System, direct, of modules: II, § 6, no. 2. System, direct, of rings (groups, fields): I, § 10, no. 3. System, free, related system: II, § 1, no. 11. System, generating, of a group: I, § 4, no. 3. System, generating, of a magma: I, § 1, no. 4. System, generating, of a module: II, § 1, no. 7. System, generating, of a projective space: II, § 9, no. 8. System, generating, of a unital magma: I, § 2, no. 1. System, generating, of an algebra: III, § 1, no. 2. System, inverse, of algebras: III, § 1, no. 6. System, inverse, of magmas: II, § 10, no. 1. System, inverse, of modules: II, § 6, no. 1. System, inverse, of rings (groups, fields): I, § 10, no. 1. System of equations of a vector subspace: II, § 7, no. 5. System of factors: III, § 2, Exercise 11. System of homogeneous coordinates of a point: II, § 9, no. 6. System of linear equations, linear system, homogeneous linear system: II, § 2, no. 8. System, trivial, of commutation factors: III, § 4, no. 7.

Table, diagonal, lower triangle, upper triangular, of matrices: II, § 10, no. 7. Table, multiplication, of an algebra: III, § 1, no. 7. Table, square, of matrices: II, § 10, no. 5. Tensor algebra of a module: III, § 5, no. 1. Tensor, contravariant, covariant tensor, mixed tensor: III, § 5, no. 6. Tensor of type (I, J): III, § 5, no. 6. Tensor, skew-symmetric: III, § 7, no. 4. Tensor, symmetric: III, § 6, no. 3. Tensorial power of a linear mapping: III, § 5, no. 2. Tensorial power of a module: III, § 5, no. 1. Term, constant, of a formal power series: III, § 2, no. 11. Term, constant, of a polynomial: III, § 2, no. 9. Term in X^α in a polynomial: III, § 2, no. 9.

Term of a formal power series: III, § 2, no. 11. Term of a polynomial: III, § 2, no. 9. Term of a sum: I, § 1, no. 2. Term of degree $p$ with respect to certain indeterminates in a formal power series: III, § 2, no. 11. Term of total degree $p$ in a formal power series: III, § 2, no. 11. Theorem, associativity: I, § 1, no. 3. Theorem, Cayley-Hamilton: III, § 8, no. 11. Theorem, commutativity: I, § 1, no. 5. Theorem, Desargues’s: II, § 9, Exercise 15. Theorem, Erdős-Kaplansky: II, § 7, Exercise 3. Theorem, fundamental, of projective geometry: II, § 9, Exercise 16. Theorem, Hall’s: I, § 6, Exercise 7. Theorem, Jordan-Hölder: I, § 4, no. 7. Theorem, Kaplansky’s: II, § 2, Exercise 2. Theorem, Krull’s: I, § 8, no. 6. Theorem, Nielsen-Schreier: I, § 7, Exercise 20. Theorem of the complete quadrilateral: II, § 9, Exercise 13. Theorem, Pappus’s: II, § 9, Exercise 14. Theorem, Schreier’s: I, § 4, no. 7. Torsion element, module, submodule: II, § 7, no. 10. Torsion-free module: II, § 7, no. 10. Total algebra of a monoid: III, § 2, no. 10. Total graduation: II, § 11, no. 1. Totally orthogonal (submodule) to a subset: II, § 2, no. 4. Trace, Cayley: III, § 2, no. 4. Trace in a quadratic algebra: III, § 2, no. 3. Trace of a matrix: II, § 10, no. 11. Trace of an element in a K-algebra: III, § 9, no. 3. Trace of an endomorphism: II, § 4, no. 3. Trace of a scalar with respect to a module: III, § 9, no. 1. Transitive operation: I, § 5, no. 5. Transitivity formulae: III, § 9, no. 4. Translation in an affine space, space of translations: II, § 9, no. 1. Translation, left, right: I, § 2, no. 2. Translation, left, right (monoid operating on itself by): I, § 5, no. 1. Transporter, strict transporter: I, § 5, no. 2. Transpose of a linear mapping, of a semi-linear mapping: II, § 2, no. 5. Transpose of a matrix: II, § 10, no. 1. Transposition: I, § 5, no. 7. Transvection: II, § 10, Exercise 11. Triangular, lower, upper triangular, matrix: II, § 10, no. 7. Trivial extension: I, § 6, no. 1.

Trivial graduation: II, § 11, no. 1. Trivial homomorphism: I, § 2, no. 1. Trivial operation: I, § 5, no. 2. Trivial solution of a homogeneous linear equation: II, § 2, no. 8. Trivial system of commutation factors: III, § 4, no. 7. Two-sided ideal: I, § 8, no. 6 and III, § 1, no. 2. Type, exponential, group of: I, § 7, Exercise 39. Type Δ, graduation of: II, § 11, no. 1.

Unimodular endomorphism: III, § 8, no. 1. Unimodular group: III, § 8, no. 9. Unimodular matrix: III, § 8, no. 3. p-unipotent element: I, § 6, Exercise 28. Unit element of an algebra: III, § 1, no. 1. Unit, left, right, in a groupoid: I, § 4, Exercise 23. Unit, unit element of a magma: I, § 2, no. 1. Unital algebra: III, § 1, no. 1. Unital algebra homomorphism, morphism: III, § 1, no. 1. Unital homomorphism: I, § 2, no. 1. Unital magma: I, § 2, no. 1. Units, matrix: II, § 10, no. 3. Universal algebra defined by a generating system related by a family of relators: III, § 2, no. 8. Universal algebra subjected to identities: III, § 2, no. 8. Universal relator: III, § 2, no. 8. Unknowns of a linear system: II, § 2, no. 8.

Value, absolute, of a rational number: I, § 9, no. 4. Value of an element of a free algebra: III, § 2, no. 8. Vandermonde determinant: III, § 8, no. 6. Variety, affine linear, affine linear variety generated by a family: II, § 9, no. 3. Variety, linear: II, § 9, nos. 3, 7 and 11. Variety, projective linear, projective linear variety generated by a family: II, § 9, nos. 7 and 11. Varieties, parallel linear: II, § 9, no. 3. Vector: II, § 1, no. 1. Vector, direction, of an affine line: II, § 9, no. 3. Vector, free, of an affine space: II, § 9, no. 1. p-vector: III, § 7, no. 1. p-vector, pure: III, § 11, no. 13. Vector rational over a subfield: II, § 8, no. 2. Vector space: II, § 1, no. 1.

Word: I, § 7, no. 2.

Weight of a homogeneous element: II, § 11, no. 1. With no fixed point (automorphism): I, § 6, Exercise 23.

Zassenhaus’s Lemma: I, § 4, no. 7. Zero: I, § 2, no. 1. Zero matrix: II, § 10, no. 22. Zero ring: I, § 8, no. 3. Zero solution of a linear equation: II, § 2, no. 8. Zero submodule: II, § 1, no. 3.