In mathematics , the commutator gives an indication of the extent to which a certain binary operation fails to be commutative . There are different definitions used in group theory and ring theory .
129-399: The commutator of two elements, g and h , of a group G , is the element This element is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by all commutators is closed and is called the derived group or
258-442: A {\displaystyle a} and b {\displaystyle b} in G {\displaystyle G} . If this additional condition holds, then the operation is said to be commutative , and the group is called an abelian group . It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation
387-414: A {\displaystyle a} and b {\displaystyle b} of G {\displaystyle G} to form an element of G {\displaystyle G} , denoted a ⋅ b {\displaystyle a\cdot b} , such that the following three requirements, known as group axioms , are satisfied: Formally,
516-412: A {\displaystyle a} and b {\displaystyle b} of a group G {\displaystyle G} , there is a unique solution x {\displaystyle x} in G {\displaystyle G} to the equation a ⋅ x = b {\displaystyle a\cdot x=b} , namely
645-430: A {\displaystyle a} and b {\displaystyle b} , the sum a + b {\displaystyle a+b} is also an integer; this closure property says that + {\displaystyle +} is a binary operation on Z {\displaystyle \mathbb {Z} } . The following properties of integer addition serve as
774-429: A − 1 ⋅ b {\displaystyle a^{-1}\cdot b} . It follows that for each a {\displaystyle a} in G {\displaystyle G} , the function G → G {\displaystyle G\to G} that maps each x {\displaystyle x} to a ⋅ x {\displaystyle a\cdot x}
903-406: A − 1 ) = φ ( a ) − 1 {\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}} for all a {\displaystyle a} in G {\displaystyle G} . However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by
1032-452: A d {\displaystyle \mathrm {ad} } is a Lie algebra homomorphism, preserving the commutator: By contrast, it is not always a ring homomorphism: usually ad x y ≠ ad x ad y {\displaystyle \operatorname {ad} _{xy}\,\neq \,\operatorname {ad} _{x}\operatorname {ad} _{y}} . The general Leibniz rule , expanding repeated derivatives of
1161-417: A d {\displaystyle \mathrm {ad} } itself as a mapping, a d : R → E n d ( R ) {\displaystyle \mathrm {ad} :R\to \mathrm {End} (R)} , where E n d ( R ) {\displaystyle \mathrm {End} (R)} is the ring of mappings from R to itself with composition as the multiplication operation. Then
1290-500: A denotes the conjugate of a by x , defined as x ax . Identity (5) is also known as the Hall–Witt identity , after Philip Hall and Ernst Witt . It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section). N.B., the above definition of the conjugate of a by x is used by some group theorists. Many other group theorists define
1419-411: A derivation on the ring R . Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation. Identities (4)–(6) can also be interpreted as Leibniz rules. Identities (7), (8) express Z - bilinearity . From identity (9), one finds that the commutator of integer powers of ring elements is: Some of the above identities can be extended to the anticommutator using
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#17329274904631548-402: A multiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted 1 {\displaystyle 1} , and the inverse of an element x {\displaystyle x} is denoted x − 1 {\displaystyle x^{-1}} . In a multiplicative group,
1677-708: A set of measure zero . The inner product of functions f and g in L ( X , μ ) is then defined as ⟨ f , g ⟩ = ∫ X f ( t ) g ( t ) ¯ d μ ( t ) {\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)} or ⟨ f , g ⟩ = ∫ X f ( t ) ¯ g ( t ) d μ ( t ) , {\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,} where
1806-548: A Hilbert space that, with the inner product induced by restriction , is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right. The sequence space l consists of all infinite sequences z = ( z 1 , z 2 , ...) of complex numbers such that the following series converges : ∑ n = 1 ∞ | z n | 2 {\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}} The inner product on l
1935-592: A definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory. Lebesgue spaces are function spaces associated to measure spaces ( X , M , μ ) , where X
2064-503: A distance function defined in this way, any inner product space is a metric space , and sometimes is known as a pre-Hilbert space . Any pre-Hilbert space that is additionally also a complete space is a Hilbert space. The completeness of H is expressed using a form of the Cauchy criterion for sequences in H : a pre-Hilbert space H is complete if every Cauchy sequence converges with respect to this norm to an element in
2193-410: A function g , the identity becomes the usual Leibniz rule for the n th derivative ∂ n ( f g ) {\displaystyle \partial ^{n}\!(fg)} . Group (mathematics) In mathematics , a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies
2322-605: A group ( G , ⋅ ) {\displaystyle (G,\cdot )} to a group ( H , ∗ ) {\displaystyle (H,*)} is a function φ : G → H {\displaystyle \varphi :G\to H} such that It would be natural to require also that φ {\displaystyle \varphi } respect identities, φ ( 1 G ) = 1 H {\displaystyle \varphi (1_{G})=1_{H}} , and inverses, φ (
2451-724: A group arose in the study of polynomial equations , starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe ) for the symmetry group of the roots of an equation, now called a Galois group . After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory —an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups , quotient groups and simple groups . In addition to their abstract properties, group theorists also study
2580-411: A group called the dihedral group of degree four, denoted D 4 {\displaystyle \mathrm {D} _{4}} . The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of
2709-467: A group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms . The set is called the underlying set of the group, and the operation is called the group operation or the group law . A group and its underlying set are thus two different mathematical objects . To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that
SECTION 20
#17329274904632838-492: A left identity (namely, e {\displaystyle e} ), and each element has a right inverse (which is e {\displaystyle e} for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However, ( G , ⋅ ) {\displaystyle (G,\cdot )}
2967-729: A left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker. In particular, assuming associativity and the existence of a left identity e {\displaystyle e} (that is, e ⋅ f = f {\displaystyle e\cdot f=f} ) and a left inverse f − 1 {\displaystyle f^{-1}} for each element f {\displaystyle f} (that is, f − 1 ⋅ f = e {\displaystyle f^{-1}\cdot f=e} ), one can show that every left inverse
3096-441: A model for the group axioms in the definition below. The integers, together with the operation + {\displaystyle +} , form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed. The axioms for a group are short and natural ... Yet somehow hidden behind these axioms
3225-972: A non-negative integer and Ω ⊂ R , the Sobolev space H (Ω) contains L functions whose weak derivatives of order up to s are also L . The inner product in H (Ω) is ⟨ f , g ⟩ = ∫ Ω f ( x ) g ¯ ( x ) d x + ∫ Ω D f ( x ) ⋅ D g ¯ ( x ) d x + ⋯ + ∫ Ω D s f ( x ) ⋅ D s g ¯ ( x ) d x {\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x} where
3354-408: A physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner . The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups. The significance of
3483-457: A point in the square to the corresponding point under the symmetry. For example, r 1 {\displaystyle r_{1}} sends a point to its rotation 90° clockwise around the square's center, and f h {\displaystyle f_{\mathrm {h} }} sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine
3612-658: A product, can be written abstractly using the adjoint representation: Replacing x {\displaystyle x} by the differentiation operator ∂ {\displaystyle \partial } , and y {\displaystyle y} by the multiplication operator m f : g ↦ f g {\displaystyle m_{f}:g\mapsto fg} , we get ad ( ∂ ) ( m f ) = m ∂ ( f ) {\displaystyle \operatorname {ad} (\partial )(m_{f})=m_{\partial (f)}} , and applying both sides to
3741-725: A real number x ⋅ y . If x and y are represented in Cartesian coordinates , then the dot product is defined by ( x 1 x 2 x 3 ) ⋅ ( y 1 y 2 y 3 ) = x 1 y 1 + x 2 y 2 + x 3 y 3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} The dot product satisfies
3870-472: A reflection along the diagonal ( f d {\displaystyle f_{\mathrm {d} }} ). Using the above symbols, highlighted in blue in the Cayley table: f h ∘ r 3 = f d . {\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.} Given this set of symmetries and the described operation,
3999-610: A ring, Hadamard's lemma applied to nested commutators gives: e A B e − A = B + [ A , B ] + 1 2 ! [ A , [ A , B ] ] + 1 3 ! [ A , [ A , [ A , B ] ] ] + ⋯ = e ad A ( B ) . {\textstyle e^{A}Be^{-A}\ =\ B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \ =\ e^{\operatorname {ad} _{A}}(B).} (For
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4128-752: A rotation over 360° which leaves the square unchanged. This is easily verified on the table. In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in D 4 {\displaystyle \mathrm {D} _{4}} , as, for example, f h ∘ r 1 = f c {\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }} but r 1 ∘ f h = f d {\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }} . In other words, D 4 {\displaystyle \mathrm {D} _{4}}
4257-574: A series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞ . {\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.} This property expresses
4386-453: A series of terms, parentheses are usually omitted. The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements e {\displaystyle e} and f {\displaystyle f} of a group are equal, because the group axioms imply e = e ⋅ f = f {\displaystyle e=e\cdot f=f} . It
4515-470: A special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces ) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations . They also form the basis of the theory of direct methods in the calculus of variations . For s
4644-521: A suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness. The second development was the Lebesgue integral , an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that
4773-444: A symmetry of the square. One of these ways is to first compose a {\displaystyle a} and b {\displaystyle b} into a single symmetry, then to compose that symmetry with c {\displaystyle c} . The other way is to first compose b {\displaystyle b} and c {\displaystyle c} , then to compose
4902-402: A uniform theory of groups started with Camille Jordan 's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by
5031-759: Is i d {\displaystyle \mathrm {id} } , as it does not change any symmetry a {\displaystyle a} when composed with it either on the left or on the right. Inverse element : Each symmetry has an inverse: i d {\displaystyle \mathrm {id} } , the reflections f h {\displaystyle f_{\mathrm {h} }} , f v {\displaystyle f_{\mathrm {v} }} , f d {\displaystyle f_{\mathrm {d} }} , f c {\displaystyle f_{\mathrm {c} }} and
5160-402: Is b ⋅ a − 1 {\displaystyle b\cdot a^{-1}} . For each a {\displaystyle a} , the function G → G {\displaystyle G\to G} that maps each x {\displaystyle x} to x ⋅ a {\displaystyle x\cdot a}
5289-621: Is countably infinite , it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable . The latter space is often in the older literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors , denoted by R , and equipped with the dot product . The dot product takes two vectors x and y , and produces
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5418-419: Is a bijection ; it is called left multiplication by a {\displaystyle a} or left translation by a {\displaystyle a} . Similarly, given a {\displaystyle a} and b {\displaystyle b} , the unique solution to x ⋅ a = b {\displaystyle x\cdot a=b}
5547-453: Is a complete metric space . A Hilbert space is a special case of a Banach space . Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert , Erhard Schmidt , and Frigyes Riesz . They are indispensable tools in the theories of partial differential equations , quantum mechanics , Fourier analysis (which includes applications to signal processing and heat transfer ), and ergodic theory (which forms
5676-781: Is a derivation on the ring R : By the Jacobi identity , it is also a derivation over the commutation operation: Composing such mappings, we get for example ad x ad y ( z ) = [ x , [ y , z ] ] {\displaystyle \operatorname {ad} _{x}\operatorname {ad} _{y}(z)=[x,[y,z]\,]} and ad x 2 ( z ) = ad x ( ad x ( z ) ) = [ x , [ x , z ] ] . {\displaystyle \operatorname {ad} _{x}^{2}\!(z)\ =\ \operatorname {ad} _{x}\!(\operatorname {ad} _{x}\!(z))\ =\ [x,[x,z]\,].} We may consider
5805-509: Is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. To say that a complex vector space H is a complex inner product space means that there is an inner product ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle } associating a complex number to each pair of elements x , y {\displaystyle x,y} of H that satisfies
5934-413: Is a bijection called right multiplication by a {\displaystyle a} or right translation by a {\displaystyle a} . The group axioms for identity and inverses may be "weakened" to assert only the existence of a left identity and left inverses . From these one-sided axioms , one can prove that the left identity is also a right identity and
6063-562: Is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: | z | = x 2 + y 2 . {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.} The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w : ⟨ z , w ⟩ = z w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This
6192-428: Is a distance function means firstly that it is symmetric in x {\displaystyle x} and y , {\displaystyle y,} secondly that the distance between x {\displaystyle x} and itself is zero, and otherwise the distance between x {\displaystyle x} and y {\displaystyle y} must be positive, and lastly that
6321-424: Is a fixed element of a ring R , identity (1) can be interpreted as a Leibniz rule for the map ad A : R → R {\displaystyle \operatorname {ad} _{A}:R\rightarrow R} given by ad A ( B ) = [ A , B ] {\displaystyle \operatorname {ad} _{A}(B)=[A,B]} . In other words, the map ad A defines
6450-1000: Is a homomorphism ψ : H → G {\displaystyle \psi :H\to G} such that ψ ∘ φ = ι G {\displaystyle \psi \circ \varphi =\iota _{G}} and φ ∘ ψ = ι H {\displaystyle \varphi \circ \psi =\iota _{H}} , that is, such that ψ ( φ ( g ) ) = g {\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g} for all g {\displaystyle g} in G {\displaystyle G} and such that φ ( ψ ( h ) ) = h {\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h} for all h {\displaystyle h} in H {\displaystyle H} . An isomorphism
6579-542: Is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups G {\displaystyle G} and H {\displaystyle H} are called isomorphic if there exists an isomorphism φ : G → H {\displaystyle \varphi :G\to H} . In this case, H {\displaystyle H} can be obtained from G {\displaystyle G} simply by renaming its elements according to
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#17329274904636708-448: Is a set, ( R , + ) {\displaystyle (\mathbb {R} ,+)} is a group, and ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} is a field . But it is common to write R {\displaystyle \mathbb {R} } to denote any of these three objects. The additive group of the field R {\displaystyle \mathbb {R} }
6837-609: Is a set, M is a σ-algebra of subsets of X , and μ is a countably additive measure on M . Let L ( X , μ ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function f in L ( X , μ ) , ∫ X | f | 2 d μ < ∞ , {\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,} and where functions are identified if and only if they differ only on
6966-443: Is also a right inverse of the same element as follows. Indeed, one has Similarly, the left identity is also a right identity: These proofs require all three axioms (associativity, existence of left identity and existence of left inverse). For a structure with a looser definition (like a semigroup ) one may have, for example, that a left identity is not necessarily a right identity. The same result can be obtained by only assuming
7095-486: Is called the weighted L space L w ([0, 1]) , and w is called the weight function. The inner product is defined by ⟨ f , g ⟩ = ∫ 0 1 f ( t ) g ( t ) ¯ w ( t ) d t . {\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.} The weighted space L w ([0, 1])
7224-627: Is complex-valued. The real part of ⟨ z , w ⟩ gives the usual two-dimensional Euclidean dot product . A second example is the space C whose elements are pairs of complex numbers z = ( z 1 , z 2 ) . Then an inner product of z with another such vector w = ( w 1 , w 2 ) is given by ⟨ z , w ⟩ = z 1 w 1 ¯ + z 2 w 2 ¯ . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.} The real part of ⟨ z , w ⟩
7353-433: Is defined by ‖ f ‖ 2 = lim r → 1 M r ( f ) . {\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.} Hardy spaces in the disc are related to Fourier series. A function f is in H ( U ) if and only if f ( z ) = ∑ n = 0 ∞
7482-457: Is defined by: ⟨ z , w ⟩ = ∑ n = 1 ∞ z n w n ¯ , {\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,} This second series converges as a consequence of the Cauchy–Schwarz inequality and
7611-564: Is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and ( H , H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,H,\langle \cdot ,\cdot \rangle )} will form a dual system . The norm is the real-valued function ‖ x ‖ = ⟨ x , x ⟩ , {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,} and
7740-590: Is identical with the Hilbert space L ([0, 1], μ ) where the measure μ of a Lebesgue-measurable set A is defined by μ ( A ) = ∫ A w ( t ) d t . {\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.} Weighted L spaces like this are frequently used to study orthogonal polynomials , because different families of orthogonal polynomials are orthogonal with respect to different weighting functions. Sobolev spaces , denoted by H or W , are Hilbert spaces. These are
7869-423: Is multiplication. More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted 0 {\displaystyle 0} , and the inverse of an element x {\displaystyle x} is denoted − x {\displaystyle -x} . Similarly, one speaks of
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#17329274904637998-440: Is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg 's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras . In the 1940s, Israel Gelfand , Mark Naimark and Irving Segal gave
8127-448: Is not a group, since it lacks a right identity. When studying sets, one uses concepts such as subset , function, and quotient by an equivalence relation . When studying groups, one uses instead subgroups , homomorphisms , and quotient groups . These are the analogues that take the group structure into account. Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A homomorphism from
8256-443: Is not abelian. The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois , extending prior work of Paolo Ruffini and Joseph-Louis Lagrange , gave a criterion for the solvability of a particular polynomial equation in terms of
8385-517: Is related to both the length (or norm ) of a vector, denoted ‖ x ‖ , and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.} Multivariable calculus in Euclidean space relies on
8514-517: Is the monster simple group , a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists. A group is a non-empty set G {\displaystyle G} together with a binary operation on G {\displaystyle G} , here denoted " ⋅ {\displaystyle \cdot } ", that combines any two elements
8643-517: Is the Laplacian and (1 − Δ) is understood in terms of the spectral mapping theorem . Apart from providing a workable definition of Sobolev spaces for non-integer s , this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators . Using these methods on a compact Riemannian manifold , one can obtain for instance
8772-521: Is the group whose underlying set is R {\displaystyle \mathbb {R} } and whose operation is addition. The multiplicative group of the field R {\displaystyle \mathbb {R} } is the group R × {\displaystyle \mathbb {R} ^{\times }} whose underlying set is the set of nonzero real numbers R ∖ { 0 } {\displaystyle \mathbb {R} \smallsetminus \{0\}} and whose operation
8901-399: Is the usual notation for composition of functions. A Cayley table lists the results of all such compositions possible. For example, rotating by 270° clockwise ( r 3 {\displaystyle r_{3}} ) and then reflecting horizontally ( f h {\displaystyle f_{\mathrm {h} }} ) is the same as performing
9030-414: Is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z ⟩ = ⟨ z , w ⟩ ¯ . {\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.} A Hilbert space
9159-497: Is then used for commutator. The anticommutator is used less often, but can be used to define Clifford algebras and Jordan algebras and in the derivation of the Dirac equation in particle physics . The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics , since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle
9288-419: Is thus customary to speak of the identity element of the group. The group axioms also imply that the inverse of each element is unique. Let a group element a {\displaystyle a} have both b {\displaystyle b} and c {\displaystyle c} as inverses. Then Therefore, it is customary to speak of the inverse of an element. Given elements
9417-479: Is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation . In phase space , equivalent commutators of function star-products are called Moyal brackets and are completely isomorphic to the Hilbert space commutator structures mentioned. The commutator has the following properties: Relation (3) is called anticommutativity , while (4) is the Jacobi identity . If A
9546-443: Is used. Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions , the operation is often function composition f ∘ g {\displaystyle f\circ g} ; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups , and automorphism groups ,
9675-417: The commutator subgroup of G . Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group . The definition of the commutator above is used throughout this article, but many group theorists define the commutator as Using the first definition, this can be expressed as [ g , h ] . Commutator identities are an important tool in group theory . The expression
9804-960: The Hodge decomposition , which is the basis of Hodge theory . The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis , whose elements are certain holomorphic functions in a complex domain. Let U denote the unit disc in the complex plane. Then the Hardy space H ( U ) is defined as the space of holomorphic functions f on U such that the means M r ( f ) = 1 2 π ∫ 0 2 π | f ( r e i θ ) | 2 d θ {\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f{\bigl (}re^{i\theta }{\bigr )}\right|^{2}\,\mathrm {d} \theta } remain bounded for r < 1 . The norm on this Hardy space
9933-703: The Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying ∫ 0 1 | f ( t ) | 2 w ( t ) d t < ∞ {\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }
10062-480: The Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis , in analogy with Cartesian coordinates in classical geometry. When this basis
10191-552: The completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers . The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus | z | , which is defined as the square root of the product of z with its complex conjugate : | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy
10320-438: The derived subgroup is central, then Rings often do not support division. Thus, the commutator of two elements a and b of a ring (or any associative algebra ) is defined differently by The commutator is zero if and only if a and b commute. In linear algebra , if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using
10449-566: The symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy . Arthur Cayley 's On the theory of groups, as depending on the symbolic equation θ n = 1 {\displaystyle \theta ^{n}=1} (1854) gives
10578-824: The triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) . {\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.} This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality , which asserts | ⟨ x , y ⟩ | ≤ ‖ x ‖ ‖ y ‖ {\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|} with equality if and only if x {\displaystyle x} and y {\displaystyle y} are linearly dependent . With
10707-423: The 180° rotation r 2 {\displaystyle r_{2}} are their own inverse, because performing them twice brings the square back to its original orientation. The rotations r 3 {\displaystyle r_{3}} and r 1 {\displaystyle r_{1}} are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields
10836-715: The Cayley table: ( f d ∘ f v ) ∘ r 2 = r 3 ∘ r 2 = r 1 f d ∘ ( f v ∘ r 2 ) = f d ∘ f h = r 1 . {\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}} Identity element : The identity element
10965-653: The ability to compute limits , and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n {\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}} consisting of vectors in R is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞ . {\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.} Just as with
11094-439: The above ± subscript notation. For example: Consider a ring or algebra in which the exponential e A = exp ( A ) = 1 + A + 1 2 ! A 2 + ⋯ {\displaystyle e^{A}=\exp(A)=1+A+{\tfrac {1}{2!}}A^{2}+\cdots } can be meaningfully defined, such as a Banach algebra or a ring of formal power series . In such
11223-420: The commutator as a Lie bracket , every associative algebra can be turned into a Lie algebra . The anticommutator of two elements a and b of a ring or associative algebra is defined by Sometimes [ a , b ] + {\displaystyle [a,b]_{+}} is used to denote anticommutator, while [ a , b ] − {\displaystyle [a,b]_{-}}
11352-507: The commutator is usually replaced by the graded commutator , defined in homogeneous components as Especially if one deals with multiple commutators in a ring R , another notation turns out to be useful. For an element x ∈ R {\displaystyle x\in R} , we define the adjoint mapping a d x : R → R {\displaystyle \mathrm {ad} _{x}:R\to R} by: This mapping
11481-499: The concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics . In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators , and measurements are orthogonal projections . The relation between quantum mechanical symmetries and unitary operators provided an impetus for
11610-405: The conjugate of a by x as xax . This is often written x a {\displaystyle {}^{x}a} . Similar identities hold for these conventions. Many identities that are true modulo certain subgroups are also used. These can be particularly useful in the study of solvable groups and nilpotent groups . For instance, in any group, second powers behave well: If
11739-440: The convergence of the previous series. Completeness of the space holds provided that whenever a series of elements from l converges absolutely (in norm), then it converges to an element of l . The proof is basic in mathematical analysis , and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space). Prior to
11868-728: The counter-diagonal ( f c {\displaystyle f_{\mathrm {c} }} ). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table. Associativity : The associativity axiom deals with composing more than two symmetries: Starting with three elements a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} of D 4 {\displaystyle \mathrm {D} _{4}} , there are two possible ways of using these three symmetries in this order to determine
11997-551: The development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists . In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers ) without necessarily identifying these elements with "geometric" vectors , such as position and momentum vectors in physical systems. Other objects studied by mathematicians at
12126-479: The development of the unitary representation theory of groups , initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space ( Koopman–von Neumann classical mechanics ) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory . The algebra of observables in quantum mechanics
12255-507: The different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group ) and of computational group theory . A theory has been developed for finite groups , which culminated with the classification of finite simple groups , completed in 2004. Since the mid-1980s, geometric group theory , which studies finitely generated groups as geometric objects, has become an active area in group theory. One of
12384-471: The distance d {\displaystyle d} between two points x , y {\displaystyle x,y} in H is defined in terms of the norm by d ( x , y ) = ‖ x − y ‖ = ⟨ x − y , x − y ⟩ . {\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.} That this function
12513-840: The dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when s is not an integer. Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If Ω is a suitable domain, then one can define the Sobolev space H (Ω) as the space of Bessel potentials ; roughly, H s ( Ω ) = { ( 1 − Δ ) − s / 2 f | f ∈ L 2 ( Ω ) } . {\displaystyle H^{s}(\Omega )=\left\{(1-\Delta )^{-s/2}f\mathrel {\Big |} f\in L^{2}(\Omega )\right\}\,.} Here Δ
12642-456: The early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators . Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from
12771-677: The existence of a right identity and a right inverse. However, only assuming the existence of a left identity and a right inverse (or vice versa) is not sufficient to define a group. For example, consider the set G = { e , f } {\displaystyle G=\{e,f\}} with the operator ⋅ {\displaystyle \cdot } satisfying e ⋅ e = f ⋅ e = e {\displaystyle e\cdot e=f\cdot e=e} and e ⋅ f = f ⋅ f = f {\displaystyle e\cdot f=f\cdot f=f} . This structure does have
12900-431: The final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the examples below illustrate. Basic facts about all groups that can be obtained directly from
13029-403: The first abstract definition of a finite group . Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein 's 1872 Erlangen program . After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded
13158-440: The first application. The result of performing first a {\displaystyle a} and then b {\displaystyle b} is written symbolically from right to left as b ∘ a {\displaystyle b\circ a} ("apply the symmetry b {\displaystyle b} after performing the symmetry a {\displaystyle a} "). This
13287-444: The following constraints: the operation is associative , it has an identity element , and every element of the set has an inverse element . Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation form an infinite group, which is generated by a single element called 1 {\displaystyle 1} (these properties characterize
13416-628: The following properties: It follows from properties 1 and 2 that a complex inner product is antilinear , also called conjugate linear , in its second argument, meaning that ⟨ x , a y 1 + b y 2 ⟩ = a ¯ ⟨ x , y 1 ⟩ + b ¯ ⟨ x , y 2 ⟩ . {\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.} A real inner product space
13545-416: The function φ {\displaystyle \varphi } ; then any statement true for G {\displaystyle G} is true for H {\displaystyle H} , provided that any specific elements mentioned in the statement are also renamed. Hilbert space In mathematics , Hilbert spaces (named after David Hilbert ) allow
13674-525: The group axioms are commonly subsumed under elementary group theory . For example, repeated applications of the associativity axiom show that the unambiguity of a ⋅ b ⋅ c = ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)} generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such
13803-602: The group axioms can be understood as follows. Binary operation : Composition is a binary operation. That is, a ∘ b {\displaystyle a\circ b} is a symmetry for any two symmetries a {\displaystyle a} and b {\displaystyle b} . For example, r 3 ∘ f h = f c , {\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },} that is, rotating 270° clockwise after reflecting horizontally equals reflecting along
13932-492: The group is the same as the set except that it has been enriched by additional structure provided by the operation. For example, consider the set of real numbers R {\displaystyle \mathbb {R} } , which has the operations of addition a + b {\displaystyle a+b} and multiplication a b {\displaystyle ab} . Formally, R {\displaystyle \mathbb {R} }
14061-424: The integers in a unique way). The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots . Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry , groups arise naturally in
14190-447: The introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt 's study of integral equations , that two square-integrable real-valued functions f and g on an interval [ a , b ] have an inner product that has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal family of functions has meaning. Schmidt exploited
14319-1183: The last expression, see Adjoint derivation below.) This formula underlies the Baker–Campbell–Hausdorff expansion of log(exp( A ) exp( B )). A similar expansion expresses the group commutator of expressions e A {\displaystyle e^{A}} (analogous to elements of a Lie group ) in terms of a series of nested commutators (Lie brackets), e A e B e − A e − B = exp ( [ A , B ] + 1 2 ! [ A + B , [ A , B ] ] + 1 3 ! ( 1 2 [ A , [ B , [ B , A ] ] ] + [ A + B , [ A + B , [ A , B ] ] ] ) + ⋯ ) . {\displaystyle e^{A}e^{B}e^{-A}e^{-B}=\exp \!\left([A,B]+{\frac {1}{2!}}[A{+}B,[A,B]]+{\frac {1}{3!}}\left({\frac {1}{2}}[A,[B,[B,A]]]+[A{+}B,[A{+}B,[A,B]]]\right)+\cdots \right).} When dealing with graded algebras ,
14448-629: The mathematical underpinning of thermodynamics ). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis . Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions , spaces of sequences , Sobolev spaces consisting of generalized functions , and Hardy spaces of holomorphic functions . Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of
14577-404: The methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional . Hilbert spaces arise naturally and frequently in mathematics and physics , typically as function spaces . Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space
14706-388: The more familiar groups is the set of integers Z = { … , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , … } {\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}} together with addition . For any two integers
14835-426: The operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, a b {\displaystyle ab} instead of a ⋅ b {\displaystyle a\cdot b} . The definition of a group does not require that a ⋅ b = b ⋅ a {\displaystyle a\cdot b=b\cdot a} for all elements
14964-407: The pioneering work of Ferdinand Georg Frobenius and William Burnside (who worked on representation theory of finite groups), Richard Brauer 's modular representation theory and Issai Schur 's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl , Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups ,
15093-406: The properties An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product . A vector space equipped with such an inner product is known as a (real) inner product space . Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it
15222-484: The requirement of respecting the group operation. The identity homomorphism of a group G {\displaystyle G} is the homomorphism ι G : G → G {\displaystyle \iota _{G}:G\to G} that maps each element of G {\displaystyle G} to itself. An inverse homomorphism of a homomorphism φ : G → H {\displaystyle \varphi :G\to H}
15351-713: The resulting symmetry with a {\displaystyle a} . These two ways must give always the same result, that is, ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) , {\displaystyle (a\circ b)\circ c=a\circ (b\circ c),} For example, ( f d ∘ f v ) ∘ r 2 = f d ∘ ( f v ∘ r 2 ) {\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})} can be checked using
15480-657: The second form (conjugation of the first element) is commonly found in the theoretical physics literature. For f and g in L , the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable . The Lebesgue spaces appear in many natural settings. The spaces L ( R ) and L ([0,1]) of square-integrable functions with respect to
15609-400: The series converges in H , in the sense that the partial sums converge to an element of H . As a complete normed space, Hilbert spaces are by definition also Banach spaces . As such they are topological vector spaces , in which topological notions like the openness and closedness of subsets are well defined . Of special importance is the notion of a closed linear subspace of
15738-576: The similarity of this inner product with the usual dot product to prove an analog of the spectral decomposition for an operator of the form where K is a continuous function symmetric in x and y . The resulting eigenfunction expansion expresses the function K as a series of the form where the functions φ n are orthogonal in the sense that ⟨ φ n , φ m ⟩ = 0 for all n ≠ m . The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in
15867-522: The space L of square Lebesgue-integrable functions is a complete metric space . As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier , Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem . Further basic results were proved in
15996-496: The space. Completeness can be characterized by the following equivalent condition: if a series of vectors ∑ k = 0 ∞ u k {\displaystyle \sum _{k=0}^{\infty }u_{k}} converges absolutely in the sense that ∑ k = 0 ∞ ‖ u k ‖ < ∞ , {\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,} then
16125-557: The study of Lie groups in 1884. The third field contributing to group theory was number theory . Certain abelian group structures had been used implicitly in Carl Friedrich Gauss 's number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker . In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers . The convergence of these various sources into
16254-551: The study of symmetries and geometric transformations : The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics . The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity . Point groups describe symmetry in molecular chemistry . The concept of
16383-946: The symbol ∘ {\displaystyle \circ } is often omitted, as for multiplicative groups. Many other variants of notation may be encountered. Two figures in the plane are congruent if one can be changed into the other using a combination of rotations , reflections , and translations . Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries . A square has eight symmetries. These are: [REDACTED] f h {\displaystyle f_{\mathrm {h} }} (horizontal reflection) [REDACTED] f d {\displaystyle f_{\mathrm {d} }} (diagonal reflection) [REDACTED] f c {\displaystyle f_{\mathrm {c} }} (counter-diagonal reflection) These symmetries are functions. Each sends
16512-425: The turn of the 20th century, in particular spaces of sequences (including series ) and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors. In the first decade of the 20th century, parallel developments led to
16641-431: Was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits . The University of Chicago 's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein , John G. Thompson and Walter Feit , laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups , with
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