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138-528: The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern , which was published in 1927 in Mathematische Annalen . Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential Moderne Algebra ,

276-477: A ′ ] Ψ , [ a ″ ] Ψ ) : ( a ′ , a ″ ) ∈ Φ } = [   ] Ψ ∘ Φ ∘ [   ] Ψ − 1 {\displaystyle \Phi /\Psi =\{([a']_{\Psi },[a'']_{\Psi }):(a',a'')\in \Phi \}=[\ ]_{\Psi }\circ \Phi \circ [\ ]_{\Psi }^{-1}}

414-581: A Realgymnasium in Nuremberg . During the 1903–1904 winter semester, she studied at the University of Göttingen , attending lectures given by astronomer Karl Schwarzschild and mathematicians Hermann Minkowski , Otto Blumenthal , Felix Klein , and David Hilbert . In 1903, restrictions on women's full enrollment in Bavarian universities were rescinded. Noether returned to Erlangen and officially reentered

552-570: A Jewish family in the Franconian town of Erlangen ; her father was the mathematician Max Noether . She originally planned to teach French and English after passing the required examinations but instead studied mathematics at the University of Erlangen , where her father lectured. After completing her doctorate in 1907 under the supervision of Paul Gordan , she worked at the Mathematical Institute of Erlangen without pay for seven years. At

690-402: A factorization system for the category . This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism f : G → H {\displaystyle f:G\rightarrow H} . The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense;

828-449: A field ) and abelian groups (modules over Z {\displaystyle \mathbb {Z} } ) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem . In the following, "module" will mean " R -module" for some fixed ring R . Let M and N be modules, and let φ  :  M  →  N be a module homomorphism . Then: In particular, if φ

966-650: A pension lodging building, after student leaders complained of living with "a Marxist-leaning Jewess". Hermann Weyl recalled that "During the wild times after the Revolution of 1918 ," Noether "sided more or less with the Social Democrats ". She was from 1919 through 1922 a member of the Independent Social Democrats , a short-lived splinter party. In the words of logician and historian Colin McLarty , "she

1104-401: A bijective correspondence between the set of subgroups of G {\displaystyle G} containing N {\displaystyle N} and the set of (all) subgroups of G / N {\displaystyle G/N} . Under this correspondence normal subgroups correspond to normal subgroups. This theorem is sometimes called the correspondence theorem ,

1242-420: A central two-volume text in the field; its second volume, published in 1931, borrowed heavily from Noether's work. Although Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin and E. Noether". Beginning in 1927, Noether worked closely with Emil Artin , Richard Brauer and Helmut Hasse on noncommutative algebras . Van der Waerden's visit

1380-495: A line – and on which PSL(3, 4) acts. One calls this Steiner system W 21 ("W" for Witt ), and then expands it to a larger Steiner system W 24 , expanding the symmetry group along the way: to the projective general linear group PGL(3, 4) , then to the projective semilinear group PΓL(3, 4) , and finally to the Mathieu group M 24 . M 24 also contains copies of PSL(2, 11) , which is maximal in M 22 , and PSL(2, 23) , which

1518-580: A noncommutative reciprocity law . This pleased her immensely. He also sent her a mathematical riddle, which he called the "m μν -riddle of syllables". She solved it immediately, but the riddle has been lost. In September of the same year, Noether delivered a plenary address ( großer Vortrag ) on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the International Congress of Mathematicians in Zürich . The congress

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1656-540: A paper about the theory of ideals in which they defined left and right ideals in a ring . The following year she published the paper Idealtheorie in Ringbereichen , analyzing ascending chain conditions with regards to (mathematical) ideals , in which she proved the Lasker–Noether theorem in its full generality. Noted algebraist Irving Kaplansky called this work "revolutionary". The publication gave rise to

1794-547: A point can be represented by [ z , 1] . Then when ad − bc ≠ 0 , the action of PGL(2, K ) is by linear transformation: In this way successive transformations can be written as right multiplication by such matrices, and matrix multiplication can be used for the group product in PGL(2, K ) . The projective special linear groups PSL( n , F q ) for a finite field F q are often written as PSL( n , q ) or L n ( q ). They are finite simple groups whenever n

1932-520: A projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of lines, preserving the incidence relation, which is exactly a collineation of a space to itself. Projective linear transforms are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transforms map planes to planes, so projective linear transforms map lines to lines), but in general not all collineations are projective linear transforms – PGL

2070-540: A return to the Soviet Union. In 1932 Emmy Noether and Emil Artin received the Ackermann–Teubner Memorial Award for their contributions to mathematics. The prize included a monetary reward of 500  ℛ︁ℳ︁ and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to

2208-418: A set of 11 points on which it acts – in fact two: the points or the lines, which corresponds to the outer automorphism – while L 2 (5) is the stabilizer of a given line, or dually of a given point. More surprisingly, the coset space L 2 (11) / ( Z  / 11 Z ), which has order 660/11 = 60 (and on which the icosahedral group acts) naturally has the structure of a buckeyball , which

2346-527: A significant change in social attitudes, including more rights for women. In 1919 the University of Göttingen allowed Noether to proceed with her habilitation (eligibility for tenure). Her oral examination was held in late May, and she successfully delivered her habilitation lecture in June 1919. Noether became a privatdozent , and she delivered that fall semester the first lectures listed under her own name. She

2484-415: A simple group (unless n = 2 and q = 2 or 3). The groups PSL(2, p ) for any prime number p were constructed by Évariste Galois in the 1830s, and were the second family of finite simple groups , after the alternating groups . Galois constructed them as fractional linear transforms, and observed that they were simple except if p was 2 or 3; this is contained in his last letter to Chevalier. In

2622-564: A subgroup of G {\displaystyle G} , and let N {\displaystyle N} be a normal subgroup of G {\displaystyle G} . Then the following hold: Technically, it is not necessary for N {\displaystyle N} to be a normal subgroup, as long as S {\displaystyle S} is a subgroup of the normalizer of N {\displaystyle N} in G {\displaystyle G} . In this case, N {\displaystyle N}

2760-476: A thesis on p-adic theory . There is no information about the first two, but it is known that Wichmann supported a student initiative that unsuccessfully attempted to revoke Noether's dismissal and died as a soldier on the Eastern Front during World War II . Noether developed a close circle of mathematicians beyond just her doctoral students who shared Noether's approach to abstract algebra and contributed to

2898-443: Is which corresponds to the order of GL( n , q ) , divided by q − 1 for projectivization; see q -analog for discussion of such formulas. Note that the degree is n − 1 , which agrees with the dimension as an algebraic group. The "O" is for big O notation , meaning "terms involving lower order". This also equals the order of SL( n , q ) ; there dividing by q − 1 is due to the determinant. The order of PSL( n , q )

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3036-581: Is a direct product decomposition of G . In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups ), left splits and right splits are equivalent by the splitting lemma , and a right split is sufficient to produce a direct sum decomposition im ⁡ κ ⊕ im ⁡ σ {\displaystyle \operatorname {im} \kappa \oplus \operatorname {im} \sigma } . In an abelian category, all monomorphisms are also normal, and

3174-480: Is a symmetric design (a set of points and an equal number of "lines", or rather blocks) such that any set of two points is contained in two lines, while any two lines intersect in two points; this is similar to a finite projective plane, except that rather than two points determining one line (and two lines determining one point), they determine two lines (respectively, points). In this case (the Paley biplane , obtained from

3312-503: Is a congruence on A / Ψ {\displaystyle A/\Psi } , and A / Φ {\displaystyle A/\Phi } is isomorphic to ( A / Ψ ) / ( Φ / Ψ ) . {\displaystyle (A/\Psi )/(\Phi /\Psi ).} Let A {\displaystyle A} be an algebra and denote Con ⁡ A {\displaystyle \operatorname {Con} A}

3450-408: Is a lattice isomorphism. Emmy Noether Amalie Emmy Noether ( US : / ˈ n ʌ t ər / , UK : / ˈ n ɜː t ə / ; German: [ˈnøːtɐ] ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra . She proved Noether's first and second theorems , which are fundamental in mathematical physics . She

3588-482: Is a map PGL(2, q ) → S q +1 . To understand these maps, it is useful to recall these facts: Thus the image is a 3-transitive subgroup of known order, which allows it to be identified. This yields the following maps: While PSL( n , q ) naturally acts on ( q − 1)/( q − 1) = 1 + q + ... + q points, non-trivial actions on fewer points are rarer. Indeed, for PSL(2, p ) acts non-trivially on p points if and only if p = 2 , 3, 5, 7, or 11; for 2 and 3

3726-436: Is an equivalence relation Φ ⊆ A × A {\displaystyle \Phi \subseteq A\times A} that forms a subalgebra of A × A {\displaystyle A\times A} considered as an algebra with componentwise operations. One can make the set of equivalence classes A / Φ {\displaystyle A/\Phi } into an algebra of

3864-440: Is an inclusion -preserving bijection between the set of subrings A {\displaystyle A} of R {\displaystyle R} that contain I {\displaystyle I} and the set of subrings of R / I {\displaystyle R/I} . Furthermore, A {\displaystyle A} (a subring containing I {\displaystyle I} )

4002-432: Is an ideal of R {\displaystyle R} if and only if A / I {\displaystyle A/I} is an ideal of R / I {\displaystyle R/I} . The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule . The isomorphism theorems for vector spaces (modules over

4140-442: Is at least 2, with two exceptions: L 2 (2), which is isomorphic to S 3 , the symmetric group on 3 letters, and is solvable ; and L 2 (3), which is isomorphic to A 4 , the alternating group on 4 letters, and is also solvable. These exceptional isomorphisms can be understood as arising from the action on the projective line . The special linear groups SL( n , q ) are thus quasisimple : perfect central extensions of

4278-488: Is best remembered for her contributions to abstract algebra . In his introduction to Noether's Collected Papers , Nathan Jacobson wrote that The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries. Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published

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4416-440: Is faithful (as the group is simple and the action is non-trivial), and yields an embedding into S p . In all but the last case, PSL(2, 11) , it corresponds to an exceptional isomorphism, where the right-most group has an obvious action on p points: Further, L 2 (7) and L 2 (11) have two inequivalent actions on p points; geometrically this is realized by the action on a biplane, which has p points and p blocks –

4554-420: Is in general a proper subgroup of the collineation group. Specifically, for n = 2 (a projective line), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line, and except for F 2 and F 3 (where PGL is the full symmetric group), PGL is a proper subgroup of the full symmetric group on these points. For n ≥ 3 , the collineation group

4692-565: Is left split (i.e., there exists some ρ : G → ker ⁡ f {\displaystyle \rho :G\rightarrow \operatorname {ker} f} such that ρ ∘ κ = id ker f {\displaystyle \rho \circ \kappa =\operatorname {id} _{{\text{ker}}f}} ), then it must also be right split, and im ⁡ κ × im ⁡ σ {\displaystyle \operatorname {im} \kappa \times \operatorname {im} \sigma }

4830-701: Is maximal in M 24 , and can be used to construct M 24 . PSL groups arise as Hurwitz groups (automorphism groups of Hurwitz surfaces – algebraic curves of maximal possibly symmetry group). The Hurwitz surface of lowest genus, the Klein quartic (genus 3), has automorphism group isomorphic to PSL(2, 7) (equivalently GL(3, 2) ), while the Hurwitz surface of second-lowest genus, the Macbeath surface (genus 7), has automorphism group isomorphic to PSL(2, 8) . In fact, many but not all simple groups arise as Hurwitz groups (including

4968-421: Is no natural notion of a projective linear transform. However, with the exception of the non-Desarguesian planes , all projective spaces are the projectivization of a linear space over a division ring though, as noted above, there are multiple choices of linear structure, namely a torsor over Gal( K  /  k ) (for n ≥ 3 ). The elements of the projective linear group can be understood as "tilting

5106-421: Is not a double cover of S 5 , but is rather a 4-fold cover. A further isomorphism is: The above exceptional isomorphisms involving the projective special linear groups are almost all of the exceptional isomorphisms between families of finite simple groups; the only other exceptional isomorphism is PSU(4, 2) ≃ PSp(4, 3), between a projective special unitary group and a projective symplectic group . Some of

5244-441: Is not a normal subgroup of G {\displaystyle G} , but N {\displaystyle N} is still a normal subgroup of the product S N {\displaystyle SN} . This theorem is sometimes called the second isomorphism theorem , diamond theorem or the parallelogram theorem . An application of the second isomorphism theorem identifies projective linear groups : for example,

5382-550: Is sometimes referred to as the third isomorphism theorem . The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem , correspondence theorem , or fourth isomorphism theorem . Let G {\displaystyle G} be a group, and N {\displaystyle N} a normal subgroup of G {\displaystyle G} . The canonical projection homomorphism G → G / N {\displaystyle G\rightarrow G/N} defines

5520-548: Is surjective then S {\displaystyle S} is isomorphic to R / ker ⁡ φ {\displaystyle R/\ker \varphi } . Let R be a ring. Let S be a subring of R , and let I be an ideal of R . Then: Let R be a ring, and I an ideal of R . Then Let I {\displaystyle I} be an ideal of R {\displaystyle R} . The correspondence A ↔ A / I {\displaystyle A\leftrightarrow A/I}

5658-513: Is surjective then N is isomorphic to M  / ker( φ ). Let M be a module, and let S and T be submodules of M . Then: Let M be a module, T a submodule of M . Let M {\displaystyle M} be a module, N {\displaystyle N} a submodule of M {\displaystyle M} . There is a bijection between the submodules of M {\displaystyle M} that contain N {\displaystyle N} and

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5796-422: Is the collineation group , which is defined axiomatically. A collineation is an invertible (or more generally one-to-one) map which sends collinear points to collinear points. One can define a projective space axiomatically in terms of an incidence structure (a set of points P , lines L , and an incidence relation I specifying which points lie on which lines) satisfying certain axioms – an automorphism of

5934-433: Is the identity matrix , and S N = GL 2 ⁡ ( C ) {\displaystyle SN=\operatorname {GL} _{2}(\mathbb {C} )} . Then the second isomorphism theorem states that: Let G {\displaystyle G} be a group, and N {\displaystyle N} a normal subgroup of G {\displaystyle G} . Then The last statement

6072-445: Is the n -dimensional vector space over a field F , namely V = F , the alternate notations PGL( n , F ) and PSL( n , F ) are also used. Note that PGL( n , F ) and PSL( n , F ) are isomorphic if and only if every element of F has an n th root in  F . As an example, note that PGL(2, C ) = PSL(2, C ) , but that PGL(2, R ) > PSL(2, R ) ; this corresponds to the real projective line being orientable, and

6210-411: Is the projective semilinear group , PΓL – this is PGL, twisted by field automorphisms ; formally, PΓL ≅ PGL ⋊ Gal( K  /  k ) , where k is the prime field for K ; this is the fundamental theorem of projective geometry . Thus for K a prime field ( F p or Q ), we have PGL = PΓL , but for K a field with non-trivial Galois automorphisms (such as F p for n ≥ 2 or C ),

6348-514: Is the order of PGL( n , q ) as above, divided by gcd( n , q − 1) . This is equal to | SZ( n , q ) | , the number of scalar matrices with determinant 1; | F  / ( F ) |, the number of classes of element that have no n th root; and it is also the number of n th roots of unity in F q . In addition to the isomorphisms there are other exceptional isomorphisms between projective special linear groups and alternating groups (these groups are all simple, as

6486-461: Is the subgroup of scalar transformations with unit determinant . Here SZ is the center of SL, and is naturally identified with the group of n th roots of unity in F (where n is the dimension of V and F is the base field ). PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups , and an element of PGL is called projective linear transformation , projective transformation or homography . If V

6624-597: Is unknown. At about the same time, Noether's father retired and her brother joined the German Army to serve in World War I . She returned to Erlangen for several weeks, mostly to care for her aging father. During her first years teaching at Göttingen, she did not have an official position and was not paid. Her lectures often were advertised under Hilbert's name, and Noether would provide "assistance". Soon after arriving at Göttingen, she demonstrated her capabilities by proving

6762-614: Is used in the construction of the buckyball surface . The group PSL(3, 4) can be used to construct the Mathieu group M 24 , one of the sporadic simple groups ; in this context, one refers to PSL(3, 4) as M 21 , though it is not properly a Mathieu group itself. One begins with the projective plane over the field with four elements, which is a Steiner system of type S(2, 5, 21) – meaning that it has 21 points, each line ("block", in Steiner terminology) has 5 points, and any 2 points determine

6900-748: The German Reichskanzler in January ;1933, Nazi activity around the country increased dramatically. At the University of Göttingen, the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by privatdozent and Noether's former student Werner Weber . Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics." Projective linear group In mathematics , especially in

7038-605: The Göttingen Gesellschaft der Wissenschaften (academy of sciences) and was never promoted to the position of Ordentlicher Professor (full professor). Noether's colleagues celebrated her fiftieth birthday, in 1932, in typical mathematicians' style. Helmut Hasse dedicated an article to her in the Mathematische Annalen , wherein he confirmed her suspicion that some aspects of noncommutative algebra are simpler than those of commutative algebra , by proving

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7176-422: The Paley digraph of order 11), the points are the affine line (the finite field) F 11 , where the first line is defined to be the five non-zero quadratic residues (points which are squares: 1, 3, 4, 5, 9), and the other lines are the affine translates of this (add a constant to all the points). L 2 (11) is then isomorphic to the subgroup of S 11 that preserve this geometry (sends lines to lines), giving

7314-544: The ascending chain condition , and objects satisfying it are named Noetherian in her honor. In the third epoch (1927–1935), she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians, even in fields far removed from her main work, such as algebraic topology . Emmy Noether

7452-503: The group theoretic area of algebra , the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P( V ). Explicitly, the projective linear group is the quotient group where GL( V ) is the general linear group of V and Z( V ) is the subgroup of all nonzero scalar transformations of V ; these are quotiented out because they act trivially on

7590-417: The invariants of finite groups . This phase marked Noether's first exposure to abstract algebra , the field to which she would make groundbreaking contributions. In Erlangen, Noether advised two doctoral students: Hans Falckenberg and Fritz Seidelmann, who defended their theses in 1911 and 1916. Despite Noether's significant role, they were both officially under the supervision of her father. Following

7728-403: The lattice of subgroups of G , while the intersection S  ∩  N is the meet . The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects. Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on

7866-404: The lattice theorem , and the fourth isomorphism theorem . The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem. The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form

8004-434: The monster group , though not all alternating groups or sporadic groups), though PSL is notable for including the smallest such groups. The groups PSL(2, Z  /  n Z ) arise in studying the modular group , PSL(2, Z ) , as quotients by reducing all elements mod n ; the kernels are called the principal congruence subgroups . A noteworthy subgroup of the projective general linear group PGL(2, Z ) (and of

8142-405: The short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from ker ⁡ f {\displaystyle \ker f} to H {\displaystyle H} and G / ker ⁡ f {\displaystyle G/\ker f} . If

8280-458: The theorem now known as Noether's theorem which shows that a conservation law is associated with any differentiable symmetry of a physical system . The paper, Invariante Variationsprobleme , was presented by a colleague, Felix Klein , on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of

8418-471: The Klein quartic (genus 3), and the order 3 biplane ( Paley biplane ) inside the buckyball surface (genus 70). The action of L 2 (11) can be seen algebraically as due to an exceptional inclusion L 2 (5) ↪ {\displaystyle \hookrightarrow } L 2 (11) – there are two conjugacy classes of subgroups of L 2 (11) that are isomorphic to L 2 (5), each with 11 elements:

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8556-430: The above maps can be seen directly in terms of the action of PSL and PGL on the associated projective line: PGL( n , q ) acts on the projective space P ( q ), which has ( q − 1)/( q − 1) points, and this yields a map from the projective linear group to the symmetric group on ( q − 1)/( q − 1) points. For n = 2 , this is the projective line P ( q ) which has ( q − 1)/( q − 1) = q + 1 points, so there

8694-402: The action of L 2 (11) by conjugation on these is an action on 11 points, and, further, the two conjugacy classes are related by an outer automorphism of L 2 (11). (The same is true for subgroups of L 2 (7) isomorphic to S 4 , and this also has a biplane geometry.) Geometrically, this action can be understood via a biplane geometry , which is defined as follows. A biplane geometry

8832-569: The action on the points and the action on the blocks are both actions on p points, but not conjugate (they have different point stabilizers); they are instead related by an outer automorphism of the group. More recently, these last three exceptional actions have been interpreted as an example of the ADE classification : these actions correspond to products (as sets, not as groups) of the groups as A 4 × Z  / 5 Z , S 4 × Z  / 7 Z , and A 5 × Z  / 11 Z , where

8970-457: The algebras A / Φ {\displaystyle A/\Phi } and im ⁡ f {\displaystyle \operatorname {im} f} are isomorphic . (Note that in the case of a group, f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} iff f ( x y − 1 ) = 1 {\displaystyle f(xy^{-1})=1} , so one recovers

9108-399: The alternating group over 5 or more letters is simple): The isomorphism L 2 (9) ≅ A 6 allows one to see the exotic outer automorphism of A 6 in terms of field automorphism and matrix operations. The isomorphism L 4 (2) ≅ A 8 is of interest in the structure of the Mathieu group M 24 . The associated extensions SL( n , q ) → PSL( n , q ) are covering groups of

9246-485: The alternating groups ( universal perfect central extensions ) for A 4 , A 5 , by uniqueness of the universal perfect central extension; for L 2 (9) ≅ A 6 , the associated extension is a perfect central extension, but not universal: there is a 3-fold covering group . The groups over F 5 have a number of exceptional isomorphisms: They can also be used to give a construction of an exotic map S 5 → S 6 , as described below. Note however that GL(2, 5)

9384-564: The arbitrary morphism f factors into ι ∘ π {\displaystyle \iota \circ \pi } , where ι is a monomorphism and π is an epimorphism (in a conormal category , all epimorphisms are normal). This is represented in the diagram by an object ker ⁡ f {\displaystyle \ker f} and a monomorphism κ : ker ⁡ f → G {\displaystyle \kappa :\ker f\rightarrow G} (kernels are always monomorphisms), which complete

9522-452: The basis for several important textbooks, such as those of van der Waerden and Deuring. Noether transmitted an infectious mathematical enthusiasm to her most dedicated students, who relished their lively conversations with her. Several of her colleagues attended her lectures and she sometimes allowed others (including her students) to receive credit for her ideas, resulting in much of her work appearing in papers not under her name. Noether

9660-519: The completion of his doctorate, Falckenberg spent time in Braunschweig and Königsberg before becoming a professor at the University of Giessen while Seidelmann became a professor in Munich . In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein . Their effort to recruit her was initially blocked by the philologists and historians among

9798-652: The development of Galois theory . Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the Russian Revolution . She was especially happy to see Soviet advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project. This attitude caused her problems in Germany, culminating in her eviction from

9936-417: The diagram may be extended by a second short exact sequence 0 → G / ker ⁡ f → H → coker ⁡ f → 0 {\displaystyle 0\rightarrow G/\operatorname {ker} f\rightarrow H\rightarrow \operatorname {coker} f\rightarrow 0} . In the second isomorphism theorem, the product SN is the join of S and N in

10074-505: The field's development, a group often referred to as the Noether school . An example of this is her close work with Wolfgang Krull , who greatly advanced commutative algebra with his Hauptidealsatz and his dimension theory for commutative rings. Another is Gottfried Köthe , who contributed to the development of the theory of hypercomplex quantities using Noether and Krull's methods. In addition to her mathematical insight, Noether

10212-491: The first abstract algebra textbook that took the groups - rings - fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke , Otto Schreier , and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem , and two laws of isomorphism when applied to groups, appear explicitly. We first present

10350-594: The former in 1911. According to her colleague Hermann Weyl and her biographer Auguste Dick , Fischer was an important influence on Noether, in particular by introducing her to the work of David Hilbert . Noether and Fischer shared lively enjoyment of mathematics and would often discuss lectures long after they were over; Noether is known to have sent postcards to Fischer continuing her train of mathematical thoughts. From 1913 to 1916, Noether published several papers extending and applying Hilbert's methods to mathematical objects such as fields of rational functions and

10488-406: The future and mathematical concepts. Noether's frugal lifestyle was at first due to her being denied pay for her work. However, even after the university began paying her a small salary in 1923, she continued to live a simple and modest life. She was paid more generously later in her life, but saved half of her salary to bequeath to her nephew, Gottfried E. Noether . Biographers suggest that she

10626-452: The group is not simple, while for 5, 7, and 11, the group is simple – further, it does not act non-trivially on fewer than p points. This was first observed by Évariste Galois in his last letter to Chevalier, 1832. This can be analyzed as follows; note that for 2 and 3 the action is not faithful (it is a non-trivial quotient, and the PSL group is not simple), while for 5, 7, and 11 the action

10764-410: The group on the complex projective line starts with setting G = GL 2 ⁡ ( C ) {\displaystyle G=\operatorname {GL} _{2}(\mathbb {C} )} , the group of invertible 2 × 2 complex matrices , S = SL 2 ⁡ ( C ) {\displaystyle S=\operatorname {SL} _{2}(\mathbb {C} )} ,

10902-540: The groups A 4 , S 4 and A 5 are the isometry groups of the Platonic solids , and correspond to E 6 , E 7 , and E 8 under the McKay correspondence . These three exceptional cases are also realized as the geometries of polyhedra (equivalently, tilings of Riemann surfaces ), respectively: the compound of five tetrahedra inside the icosahedron (sphere, genus 0), the order 2 biplane (complementary Fano plane ) inside

11040-649: The handkerchief from her blouse and ignored the increasing disarray of her hair during a lecture. Two female students once approached her during a break in a two-hour class to express their concern, but they were unable to break through the energetic mathematical discussion she was having with other students. Noether did not follow a lesson plan for her lectures. She spoke quickly and her lectures were considered difficult to follow by many, including Carl Ludwig Siegel and Paul Dubreil . Students who disliked her style often felt alienated. "Outsiders" who occasionally visited Noether's lectures usually spent only half an hour in

11178-415: The higher "ordinary" professorship, which was a civil-service position. Although it recognized the importance of her work, the position still provided no salary. Noether was not paid for her lectures until she was appointed to the special position of Lehrbeauftragte für Algebra a year later. Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she

11316-402: The hyperplane are proper projective maps, and accounts for the remaining n dimensions. As for Möbius transformations , the group PGL(2, K ) can be interpreted as fractional linear transformations with coefficients in K . Points in the projective line over K correspond to pairs from K , with two pairs being equivalent when they are proportional. When the second coordinate is non-zero,

11454-403: The image of f {\displaystyle f} is a subalgebra of B {\displaystyle B} , the relation given by Φ : f ( x ) = f ( y ) {\displaystyle \Phi :f(x)=f(y)} (i.e. the kernel of f {\displaystyle f} ) is a congruence on A {\displaystyle A} , and

11592-421: The isomorphism theorems of the groups . Let G and H be groups, and let f  :  G  →  H be a homomorphism . Then: In particular, if f is surjective then H is isomorphic to G  / ker( f ). This theorem is usually called the first isomorphism theorem . Let G {\displaystyle G} be a group. Let S {\displaystyle S} be

11730-461: The latter died of tuberculosis shortly before his defense. Grell defended his thesis in 1926 and went on to work at the University of Jena and the University of Halle , before losing his teaching license in 1935 due to accusations of homosexual acts. He was later reinstated and became a professor at Humboldt University in 1948. Noether then supervised Werner Weber and Jakob Levitzki , who both defended their theses in 1929. Weber, who

11868-410: The lattice of submodules of M / N {\displaystyle M/N} and the lattice of submodules of M {\displaystyle M} that contain N {\displaystyle N} ). To generalise this to universal algebra , normal subgroups need to be replaced by congruence relations . A congruence on an algebra A {\displaystyle A}

12006-494: The linear (vector space) structure", the projective linear group is defined constructively, as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure". This is reflected in the notation: PGL( n , F ) is the group associated to GL( n , F ) , and is the projective linear group of ( n − 1) -dimensional projective space, not n -dimensional projective space. A related group

12144-466: The masculine German article. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship to Princeton University for the 1927–1928 academic year from the Rockefeller Foundation . In Göttingen, Noether supervised more than a dozen doctoral students, though most were together with Edmund Landau and others as she

12282-479: The most important mathematical theorems ever proved in guiding the development of modern physics". In the second epoch (1920–1926), she began work that "changed the face of [abstract] algebra". In her classic 1921 paper Idealtheorie in Ringbereichen ( Theory of Ideals in Ring Domains ), Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications. She made elegant use of

12420-505: The most promising of Noether's students, was awarded his doctorate in 1930. He worked in Hamburg, Marden and Göttingen and is known for his contributions to arithmetic geometry . Fitting graduated in 1931 with a thesis on abelian groups and is remembered for his work in group theory , particularly Fitting's theorem and the Fitting lemma . He died at the age of 31 from a bone disease. Witt

12558-455: The natural action of GL( V ) on V descends to an action of PGL( V ) on the projective space P ( V ). The projective linear groups therefore generalise the case PGL(2, C ) of Möbius transformations (sometimes called the Möbius group ), which acts on the projective line . Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving

12696-398: The notion of a normal subgroup replaced by the notion of an ideal . Let R {\displaystyle R} and S {\displaystyle S} be rings, and let φ : R → S {\displaystyle \varphi :R\rightarrow S} be a ring homomorphism . Then: In particular, if φ {\displaystyle \varphi }

12834-524: The notion of kernel used in group theory in this case.) Given an algebra A {\displaystyle A} , a subalgebra B {\displaystyle B} of A {\displaystyle A} , and a congruence Φ {\displaystyle \Phi } on A {\displaystyle A} , let Φ B = Φ ∩ ( B × B ) {\displaystyle \Phi _{B}=\Phi \cap (B\times B)} be

12972-461: The numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules. It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem , as one of isomorphism theorems, but when included, it is the last one. The statements of the theorems for rings are similar, with

13110-537: The origin (or placing a camera at the origin), and turning one's angle of view, then projecting onto a flat plane. Rotations in axes perpendicular to the hyperplane preserve the hyperplane and yield a rotation of the hyperplane (an element of SO( n ), which has dimension ⁠ 1 + 2 + ⋯ + ( n − 1 ) = ( n 2 ) {\displaystyle \textstyle {1+2+\cdots +(n-1)={\binom {n}{2}}}} ⁠ .), while rotations in axes parallel to

13248-407: The philosophical faculty, who insisted that women should not become privatdozenten . In a joint department meeting on the matter, one faculty member protested: "What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?" Hilbert, who believed Noether's qualifications were the only important issue and that the sex of the candidate

13386-602: The plane" along one of the axes, and then projecting to the original plane, and also have dimension n . A more familiar geometric way to understand the projective transforms is via projective rotations (the elements of PSO( n + 1) ), which corresponds to the stereographic projection of rotations of the unit hypersphere, and has dimension ⁠ 1 + 2 + ⋯ + n = ( n + 1 2 ) {\displaystyle \textstyle {1+2+\cdots +n={\binom {n+1}{2}}}} ⁠ . Visually, this corresponds to standing at

13524-466: The projective linear group is a proper subgroup of the collineation group, which can be thought of as "transforms preserving a projective semi -linear structure". Correspondingly, the quotient group PΓL / PGL = Gal( K  /  k ) corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure. One may also define collineation groups for axiomatically defined projective spaces, where there

13662-406: The projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group. The projective special linear group , PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: where SL( V ) is the special linear group over V and SZ( V )

13800-426: The projective special linear group PSL(2, Z [ i ]) ) is the symmetries of the set {0, 1, ∞} ⊂ P ( C ) which is known as the anharmonic group , and arises as the symmetries of the six cross-ratios . The subgroup can be expressed as fractional linear transformations , or represented (non-uniquely) by matrices, as: Note that the top row is the identity and the two 3-cycles, and are orientation-preserving, forming

13938-431: The projective special linear group only being the orientation-preserving transformations. PGL and PSL can also be defined over a ring , with an important example being the modular group , PSL(2, Z ) . The name comes from projective geometry , where the projective group acting on homogeneous coordinates ( x 0  : x 1  : ... : x n ) is the underlying group of the geometry. Stated differently,

14076-463: The rank of Privatdozent . Noether remained a leading member of the Göttingen mathematics department until 1933; her students were sometimes called the "Noether Boys". In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas; her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra . By

14214-403: The room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out." She used her lectures as a spontaneous discussion time with her students, to think through and clarify important problems in mathematics. Some of her most important results were developed in these lectures, and the lecture notes of her students formed

14352-492: The same letter and attached manuscripts, Galois also constructed the general linear group over a prime field , GL( ν , p ) , in studying the Galois group of the general equation of degree p . The groups PSL( n , q ) (general n , general finite field) for any prime power q were then constructed in the classic 1870 text by Camille Jordan , Traité des substitutions et des équations algébriques . The order of PGL( n , q )

14490-516: The same time, she lectured and performed research at the Institute for Advanced Study in Princeton, New Jersey . Noether's mathematical work has been divided into three " epochs ". In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields . Her work on differential invariants in the calculus of variations , Noether's theorem , has been called "one of

14628-427: The same type by defining the operations via representatives; this will be well-defined since Φ {\displaystyle \Phi } is a subalgebra of A × A {\displaystyle A\times A} . The resulting structure is the quotient algebra . Let f : A → B {\displaystyle f:A\rightarrow B} be an algebra homomorphism . Then

14766-481: The sequence is right split (i.e., there is a morphism σ that maps G / ker ⁡ f {\displaystyle G/\operatorname {ker} f} to a π -preimage of itself), then G is the semidirect product of the normal subgroup im ⁡ κ {\displaystyle \operatorname {im} \kappa } and the subgroup im ⁡ σ {\displaystyle \operatorname {im} \sigma } . If it

14904-560: The set of all congruences on A {\displaystyle A} . The set Con ⁡ A {\displaystyle \operatorname {Con} A} is a complete lattice ordered by inclusion. If Φ ∈ Con ⁡ A {\displaystyle \Phi \in \operatorname {Con} A} is a congruence and we denote by [ Φ , A × A ] ⊆ Con ⁡ A {\displaystyle \left[\Phi ,A\times A\right]\subseteq \operatorname {Con} A}

15042-762: The set of all congruences that contain Φ {\displaystyle \Phi } (i.e. [ Φ , A × A ] {\displaystyle \left[\Phi ,A\times A\right]} is a principal filter in Con ⁡ A {\displaystyle \operatorname {Con} A} , moreover it is a sublattice), then the map α : [ Φ , A × A ] → Con ⁡ ( A / Φ ) , Ψ ↦ Ψ / Φ {\displaystyle \alpha :\left[\Phi ,A\times A\right]\to \operatorname {Con} (A/\Phi ),\Psi \mapsto \Psi /\Phi }

15180-577: The society. American physicists Leon M. Lederman and Christopher T. Hill argue in their book Symmetry and the Beautiful Universe that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics , possibly on a par with the Pythagorean theorem ". When World War I ended, the German Revolution of 1918–1919 brought

15318-470: The spring of 1900, she took the examination for teachers of these languages and received an overall score of sehr gut (very good). Her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen , at which her father was a professor. This was an unconventional decision; two years earlier, the Academic Senate of

15456-656: The subgroup of determinant 1 matrices, and N {\displaystyle N} the normal subgroup of scalar matrices C × I = { ( a 0 0 a ) : a ∈ C × } {\displaystyle \mathbb {C} ^{\times }\!I=\left\{\left({\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right):a\in \mathbb {C} ^{\times }\right\}} , we have S ∩ N = { ± I } {\displaystyle S\cap N=\{\pm I\}} , where I {\displaystyle I}

15594-415: The submodules of M / N {\displaystyle M/N} . The correspondence is given by A ↔ A / N {\displaystyle A\leftrightarrow A/N} for all A ⊇ N {\displaystyle A\supseteq N} . This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between

15732-446: The term Noetherian for objects which satisfy the ascending chain condition. In 1924, a young Dutch mathematician, Bartel Leendert van der Waerden , arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison". After returning to Amsterdam, he wrote Moderne Algebra ,

15870-603: The time of her plenary address at the 1932 International Congress of Mathematicians in Zürich , her algebraic acumen was recognized around the world. The following year, Germany's Nazi government dismissed Jews from university positions , and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania . There, she taught graduate and post-doctoral women including Marie Johanna Weiss , Ruth Stauffer, Grace Shover Quinn, and Olga Taussky-Todd . At

16008-409: The time, women were largely excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen , a world-renowned center of mathematical research. The philosophical faculty objected, however, and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919, allowing her to obtain

16146-922: The trace of Φ {\displaystyle \Phi } in B {\displaystyle B} and [ B ] Φ = { K ∈ A / Φ : K ∩ B ≠ ∅ } {\displaystyle [B]^{\Phi }=\{K\in A/\Phi :K\cap B\neq \emptyset \}} the collection of equivalence classes that intersect B {\displaystyle B} . Then Let A {\displaystyle A} be an algebra and Φ , Ψ {\displaystyle \Phi ,\Psi } two congruence relations on A {\displaystyle A} such that Ψ ⊆ Φ {\displaystyle \Psi \subseteq \Phi } . Then Φ / Ψ = { ( [

16284-412: The university had declared that allowing mixed-sex education would "overthrow all academic order". One of just two women in a university of 986 students, Noether was allowed only to audit classes rather than participate fully, and she required the permission of individual professors whose lectures she wished to attend. Despite these obstacles, on 14 July 1903, she passed the graduation exam at

16422-503: The university in October 1904, declaring her intention to focus solely on mathematics. She was one of six women in her year (two auditors) and the only woman in her chosen school. Under the supervision of Paul Gordan , she wrote her dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form ( On Complete Systems of Invariants for Ternary Biquadratic Forms ), in 1907, graduating summa cum laude later that year. Gordan

16560-422: The works of her students above all. Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and lectured at a local coffee house. Later, after Nazi Germany dismissed her from teaching, she invited students into her home to discuss their plans for

16698-465: Was near-sighted and talked with a minor lisp during her childhood. A family friend recounted a story years later about young Noether quickly solving a brain teaser at a children's party, showing logical acumen at an early age. She was taught to cook and clean, as were most girls of the time, and took piano lessons. She pursued none of these activities with passion, although she loved to dance. She had three younger brothers. The eldest, Alfred Noether,

16836-420: Was a member of the "computational" school of invariant researchers, and Noether's thesis ended with a list of over 300 explicitly worked-out invariants. This approach to invariants was later superseded by the more abstract and general approach pioneered by Hilbert. Although it had been well received, Noether later described her thesis and some subsequent similar papers she produced as "crap". All of her later work

16974-437: Was attended by 800 people, including Noether's colleagues Hermann Weyl , Edmund Landau , and Wolfgang Krull . There were 420 official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career. When Adolf Hitler became

17112-514: Was born in 1883 and was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether was born in 1884, studied in Munich and made contributions to applied mathematics . He was executed in the Soviet Union in 1941. The youngest, Gustav Robert Noether, was born in 1889. Very little is known about his life; he suffered from chronic illness and died in 1928. Noether showed early proficiency in French and English. In

17250-436: Was born on 23 March 1882. She was the first of four children of mathematician Max Noether and Ida Amalia Kaufmann, both from Jewish merchant families. Her first name was "Amalie", but she began using her middle name at a young age and she invariably used the name "Emmy Noether" in her adult life and her publications. In her youth, Noether did not stand out academically although she was known for being clever and friendly. She

17388-671: Was considered only a modest mathematician, would later take part in driving Jewish mathematicians out of Göttingen. Levitzki worked first at Yale University and then at the Hebrew University of Jerusalem in Palestine, making significant contributions (in particular Levitzky's theorem and the Hopkins–Levitzki theorem ) to ring theory . Other Noether Boys included Max Deuring , Hans Fitting , Ernst Witt , Chiungtze C. Tsen and Otto Schilling . Deuring, who had been considered

17526-407: Was described by Pavel Alexandrov , Albert Einstein , Jean Dieudonné , Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics . As one of the leading mathematicians of her time, she developed theories of rings , fields , and algebras . In physics, Noether's theorem explains the connection between symmetry and conservation laws . Noether was born to

17664-474: Was forced to find a new advisor due to Noether's emigration. Under Helmut Hasse , he completed his PhD in 1934 at the University of Marburg . He later worked as a post doc at Trinity College, Cambridge , before moving to the United States. Noether's other students were Wilhelm Dörnte, who received his doctorate in 1927 with a thesis on groups, Werner Vorbeck, who did so in 1935 with a thesis on splitting fields , and Wolfgang Wichmann, who did so 1936 with

17802-687: Was in a completely different field. From 1908 to 1915, Noether taught at Erlangen's Mathematical Institute without pay, occasionally substituting for her father, Max Noether , when he was too ill to lecture. She joined the Circolo Matematico di Palermo in 1908 and the Deutsche Mathematiker Vereinigung in 1909. In 1910 and 1911, she published an extension of her thesis work from three variables to n variables. Gordan retired in 1910, and Noether taught under his successors, Erhard Schmidt and Ernst Fischer , who took over from

17940-774: Was initially supervised by Noether, but her position was revoked in April 1933 and he was assigned to Gustav Herglotz instead. He received his PhD in July 1933 with a thesis on the Riemann-Roch theorem and zeta-functions , and went on to make several contributions that now bear his name . Tsen, best remembered for proving Tsen's theorem , received his doctorate in December of the same year. He returned to China in 1935 and started teaching at National Chekiang University , but died only five years later. Schilling also began studying under Noether but

18078-705: Was irrelevant, objected with indignation and scolded those protesting her habilitation. Although his exact words have not been preserved, his objection is often said to have included the remark that the university was "not a bathhouse." According to Pavel Alexandrov 's recollection, faculty members' opposition to Noether was based not just in sexism, but also in their objections to her social-democratic political beliefs and Jewish ancestry. Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. She had previously received medical care for an eye condition, but its nature and impact on her death

18216-416: Was mostly unconcerned about appearance and manners, focusing on her studies. Olga Taussky-Todd , a distinguished algebraist taught by Noether, described a luncheon during which Noether, wholly engrossed in a discussion of mathematics, "gesticulated wildly" as she ate and "spilled her food constantly and wiped it off from her dress, completely unperturbed". Appearance-conscious students cringed as she retrieved

18354-477: Was not a Bolshevist but was not afraid to be called one." Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933, he tried to help her gain a chair at Moscow State University through the Soviet Education Ministry . Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for

18492-531: Was not allowed to supervise dissertations on her own. Her first was Grete Hermann , who defended her dissertation in February ;1925. Although she is best remembered for her work on the foundations of quantum mechanics , her dissertation was considered an important contribution to ideal theory . Hermann later spoke reverently of her "dissertation-mother". Around the same time, Heinrich Grell and Rudolf Hölzer wrote their dissertations under Noether, though

18630-430: Was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were the first of several in 1923. Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends. He dubbed her der Noether , using der as an epithet rather than as

18768-448: Was recorded as having given at least five semester-long courses at Göttingen: In the winter of 1928–1929, Noether accepted an invitation to Moscow State University , where she continued working with P.S. Alexandrov . In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry . She worked with the topologists Lev Pontryagin and Nikolai Chebotaryov , who later praised her contributions to

18906-534: Was respected for her consideration of others. Although she sometimes acted rudely toward those who disagreed with her, she nevertheless gained a reputation for constant helpfulness and patient guidance of new students. Her loyalty to mathematical precision caused one colleague to name her "a severe critic", but she combined this demand for accuracy with a nurturing attitude. In Noether's obituary, Van der Waerden described her as Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted

19044-474: Was still not paid for her work. Three years later, she received a letter from Otto Boelitz  [ de ] , the Prussian Minister for Science, Art, and Public Education, in which he conferred on her the title of nicht beamteter ausserordentlicher Professor (an untenured professor with limited internal administrative rights and functions). This was an unpaid "extraordinary" professorship , not

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