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15

From Pais's book Inward Bound, chapter 13: Wigner had become interested in $n>2$ identical particle problem. He rapidly mastered the case $n=3$ (without spin). His methods were rather laborious; for example, he had to solve a (reducible) equation of degree six. It would be pretty awful to go on this way to higher $n$. So, Wigner told me, he went to ...

12

Galois's introduced the term group in 1831 in a paper on solutions of polynomials by radicals, but he only considered permutation groups. Without an explicit concept group properties of permutations were used by Lagrange, Ruffini and Cauchy before him. In particular in Lagrange's Reflexions sur la Resolution Algebrique des Equation (1770-71) a theorem on ...

11

The group concept (and all other important concepts) evolved gradually. Galois did not define the group as we know it now. For him, a group was a group of substitutions on a finite set. Substitutions on finite sets were considered by Cauchy and Lagrange before Galois. Including premutations of the roots of algebraic equations. Galois contribution is that he ...

10

The quoted 1946 English edition of Pontryagin's 1938 book is not the first appearance of kernel. It's already in the first English edition (1939). It's earlier in e.g. 1938 papers of J. H. C. Whitehead (p. 703) and Montgomery-Zippin (p. 364). The original German Kern is in P. Alexandroff and H. Hopf, Topologie I (1935, p. 557), and as Jan Peter comments ...

8

It came to physics a bit earlier than quantum mechanics. The homomorphism $SU(2)\to SO(3)$ was discovered by Cayley (1843), Hamilton (1847), and Klein (1875) in their pure mathematical studies, and came to the attention of physicists through the theory of rigid body rotation (classical mechanics). It was Klein who brought it to the attention of physicists. ...

8

Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt: "Invariant integration on one or another special class of groups has long been known and used. A detailed computation of the invariant integral on $\mathfrak{SD}(n)$ was given in 1897 by HURWITZ [1]. ...

8

No, he did not. You are used to see Galois theory from the modern point of view, developed by Emmy Noether and two of her students: van der Waerden and Emil Artin. I suggest that you read B. Melvin Kiernan's The Development of Galois Theory from Lagrange to Artin (Archive for History of Exact Sciences Vol. 8, No. 1/2, pp. 40–154).

6

The center (originally the central) seems to have appeared between the first and second edition of Burnside’s book (1897, §53 vs. 1911, §93) and more precisely in de Séguier (1904, §51): Ainsi l’ensemble des éléments normaux de $\mathrm G$ est un diviseur normal qui sera dit central de $\mathrm G$. (Jahrbuch: “Die Gesamtheit invarianter Elemente einer ...

6

See the end of the Wikipedia link in your first sentence. The source is Rotman's Introduction to the Theory of Groups (1995), which reads on p.383:"There are today several different proofs of this theorem, some "algebraic" and some "geometric". The first geometric proof was given by Baer and Levi in 1936, and this is the proof we present. There is another ...

6

According to an entry attributed to John Aldrich in Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, the word "kernel" in the algebraic sense was first used in the 1946 translation of Pontryagin's book Topological Groups by Emma Lehmer. So kernels of group homomorphisms came first. An unrelated use as in "integral kernel" occurs earlier,...

6

A first general treatment (that is, with an abstract notion of field, which is how I understand the question) of Galois theory was given by Heinrich Weber in "Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie" Mathematische Annalen 43 (1893) 548 - 549

5

The article The Abstract Group Concept, from the McTutor archive, gives an accounting of the steps towards the modern abstract definition. In brief, Cayley made the first stumbling attempts (citing the associative law explicitly) in an 1854 paper, but not until 1895 did Weber give the modern definition, in his Lehrbuch der Algebra. Weber included infinite ...

5

The traditional answer to questions on symmetry would be to point to Felix Klein's Erlangen Program as a way of systematizing the study of symmetry by focusing on the symmetries of a manifold as the essential feature thereof. Earlier sources include of course Galois' group theory, which can always be interpreted as a study of symmetry. In a multicultural ...

4

Try these references: Section 7.5 of History of Topology, edited by I. M. James. Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology, volume 3, edited by C.E. All and R. Lowen

4

In the 17th century Bachet and Fermat gave algebraic formulas for doubling a point on a cubic, and Newton showed how to do it in terms of chords and tangents. But that is as far as geometry progressed on its own, from there the path did not go from geometry to the group law, but the other way. In 1834 Jacobi pointed out a possible connection between cubic ...

4

In my opinion, there are two separate questions being asked here. The expectation that the answer should concern 18th-19th century was not formulated in the statement of the question(s), although one can say that this is when mathematicians started to  focus their attention on the concept of symmetry." At any rate, imposing such a time frame leaves out ...

3

The earliest occurence Milnor mentions in his survey Differential Topology Forty-six Years Later is Whitehead's paper On the Homotopy Groups of Spheres and Rotation Groups (1942). If $J_n$ is the image of the stable Whitehead homomorphism from the $n$th stable homotopy group of rotation groups to the $n$th stable homotopy group of spheres, then $J_n$ is ...

2

"Almost everything" was found before the general modern definition of a group was extablished:-) I am not sure who gave the first definition of the abstract group (as a set with an operation satisfying such and such axioms). But probably this happened in 20-s century (various people are credited with this). For 19 century mathematicians a group was a group ...

2

The he earliest one seems to be C. Jordan, Traite des substitutions algebriques: C. Jordan, Traite des substitutions et des equations algebriques. (French) Paris. Gauthier-Villars. 1870. Published: 1870 The most comprehensive one covering this period is Dickson, Linear groups with an exposition of the Galois field theory. (German) Leipzig: B. G. Teubner. ...

2

It is true that Frobenius did not have the notion of a group representation in his celebrated "trilogy" of 1896 papers on factorization of the group determinant problem, posed to him by Dedekind in their correspondence the same year. Instead, he defined characters by working with a commutative complex algebra (which he later identified as the center of the ...

2

Mulliken credits Georg Placzek in his autobiography (1989, p. 90). According to T. Oka (2011)(pdf): Placzek (1934) introduced the currently used symbols of irreducible representations such as A (symmetric), B (antisymmetric), E (“entartet”, doubly degenerate), F (triply degenerate), subscript 1 and 2 (to specify symmetry with respect to C2) and ...

2

Early examples are Burnside (1910, pp. 324-325; 1911, p. 271) where $\color{red}{\textrm{ir}}$reducible representations are called $\Gamma$, $\Gamma_1$, $\Gamma_2$, etc. (Earlier in (1901) he had called them $G_1$, $G_2$, etc.) Speiser (1923, p. 104; 1927, p. 151) uses the same convention, but allows $\Gamma$ to be reducible. Note that for them a ...

2

On my opinion, there are two things which triggered this process in 19th cetury: Galois theory (and the work of his predecessors, like Lagrange and Cauchy). They introduced the notion of group. Work on crystallography (second part of 19th century) where groups are explicitly used to develop a physical theory. Of course the study of symmetry is much older ...

1

I found this statement in a short IAS article, In [Conway's] view, conformal field theory is too complicated to understand, and thus too complicated to be the only answer. However, seeing as Conway's work on the Monster group and 'moonshine' seems to predate its proposed application to string theory by a number of years, I'm a bit skeptical that he ...

1

As a side remark: I recommend warmly the book: "From Summetria to Symmetry: The Making of a Revolutionary Scientific Concept" by Hon\Goldstein, which deals with the emergence of the concept of symmetry till the beginning of the 19th cent.

1

The earliest applications of groups in geometry are dealing with continuous groups (Lie, Klein 1872). You seem to be talking about discrete groups. Their earliest application in a geometric subject was in crystallography in the 1879-1900. See "Crystallographic groups" in Wikipedia.

1

Maybe Dedekind could have done what you are looking for. The sources for the early development apparently include Cauchy, Galois, Abel, Cayley, etc. However, according to Bell "Men of Mathematics" (enjoyable to read, but not a really scholarly work), page 282: "This abstract point of view is that now current. It was not Cauchy's, but was introduced by ...

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