Questions tagged [group-theory]

The study of algebraic structures known as groups.

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Intuitions for Frobenius' generalization of characters to nonabelian finite group given the historical context

I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius. According to most of the related papers (e.g. Pioneers of ...
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Why is it called a group action?

A group action has two laws which roughly correspond to associativity and identity $ \phi : (G : \textrm{Group}) \times (S : \textrm{Set}) \rightarrow S \\ \forall a, b : G . \forall c : S. \phi(a,\...
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A Survey of Modern Algebra, 1st Edition (Birkhoff and Mac Lane): The Direct Product

Background information: I recently asked a question about the history of the concepts of the direct sum and the tensor product in group and module theory, and was given a very concise and thorough ...
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6 votes
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First Use of the Short Exact Sequence

A question I've been curious about for a long period of time and tried to find the answer to myself a number of times (but apparently never been able to figure out just the right thing to type into ...
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2 votes
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Continuation on Galois’ lost memoir(s): was Poisson right in his review of Galois’ memoir?

A recent question asked whether Galois’ “lost” memoir has been since recovered (similar to Abel’s lost memoir). The memoir referenced in the question is the one Galois submitted to the French “...
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Why is the number of elements in a group called "order"?

This is a question that I have for a long time, Maybe it is something silly, but I really want to know. Why is the number of elements in a group called "order"? I mean, the word "order&...
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History of PSL(2,Z)

I want to know the mathematician who named PSL(2,Z) as the modular group first. Also, I would be pleased if someone suggest to me some papers about the history of the modular group. Thanks a lot in ...
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Origin of the recurrence relation for Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients $C_{\pm }(J,M)$ arise in quantum mechanics in the problem of addition of angular momentum. They also arise in mathematics in the more theoretical (but related) problem ...
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Representation theory in physics

The idea of using Lie groups in physics can be easily understood intuitively, but what are the origins of the use of representation theory of Lie groups and Lie algebras in physics? We mathematicians ...
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Lagrange's Theorem as he stated it

In Wikipedia, I found that Lagrange did not state Lagrange's Theorem in its general form. He stated "If a Polynomial in $n$ variables has its variables permuted in all $n!$ ways, the number of ...
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Origin of the term 'index of a subgroup'

The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$. Why did someone decide to call this an 'index'?
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First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
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Exact quote (and source of) by John Conway regarding the action/origin of the Monster?

I vaguely remember reading a quote/listening to a statement by John Conway which I can paraphrase as follows: "I do not know what object the Monster group acts on, but it seems to exist because ...
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Why are faithful actions called faithful and who first called them faithful?

This is a cross post from MSE I want to know why are faithful actions called faithful and who first called them faithful? Definition: An action $G$ on $X$ is faithful when ${g_1 \neq g_2 \Rightarrow ...
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Any idea on how Lagrange came up with similar functions concept in (proto)group theory?

Lagrange defines "similar functions" as functions of the roots of an equation where they change values only at the same kind of permutations of the roots. What's a possible predecessor of the idea of ...
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Writing functions on the right

In group theory, writing functions on the right is a common, though not universal practice. Thus, given mappings $f$, $g$ and group element $\alpha$, one might write $\alpha f$ and $\alpha (f \circ ...
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How did $SU(2)$ came into physics?

It is natural for physicists to consider the group $SO(3)$. Presumably, $SU(2)$ came into physics because of quantum mechanics. How did people realize that when studying rotation of a physical system, ...
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On the history of Haar measure

Haar measure is a well-known concept in measure theory. Many books are perfectly dedicated to present its existence and uniqueness such as measure theory for D. Cohn. I am looking for a good ...
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1 answer
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Did Galois make use of the concept of a basis?

I've been reading Galois' First Memoir, where he introduces Galois Theory by giving a sufficient and necessary condition for a polynomial to be solvable by radicals. The proofs are a bit sketchy and ...
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How did the terms "center" and "centralizer" come up in group theory?

Usually the word center means the center of a circle. I have encountered the word center in group theory, but do not see any connection with the center of a circle. I think the history of group theory ...
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What does the "G" for the similitude groups stand for?

When we have a bilinear symmetric/ bilinear anti-symmetric/hermitian form $b$ on a real/complex vector space $V$, one can consider the group of invertible matrices $A \in GL(V)$ which respect $b$, ...
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History of group theory character tables (as used in physics and chemistry)

Does anyone know who started using the symbols A, B, E, T (First column, left) for showing irreducible representations of symmetry groups? In older maths books I see capital gamma. Herein A= totally ...
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Earliest known usage of letter gamma "Γ" for reducible representation in group theory

Does any know the earliest known usage of the Greek letter gamma for showing a reducible representation of a group? This symbolism is commonly used in character tables in chemical applications of ...
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14 votes
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A basic mistake by Cayley

Arthur Cayley's first paper on abstract groups, in 1854, can be found in his Collected Papers on the Internet Archive, starting at https://archive.org/stream/collectedmathema02cayluoft#page/122/mode/...
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Whence “homomorphism”, “homomorphic”?

The kernel question leads to another : Today, homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” ...
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Who first identified the group structure of an elliptic curve?

I find it amazing that the geometric construction that underlies the group law for elliptic curves gives rise to a group law. Q: Who was the first to identify the group law for elliptic curves and, ...
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What are the origins of the study of symmetry as a subject in itself?

Symmetry has become a central concept in mathematics. The Euclidean concept of similarity is an example of symmetry, but similarity was not a subject of study in itself. Q: How did symmetry come to ...
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Early discoveries combining groups and geometry?

More specifically: When were the symmetries of polygons/solids first presented as groups in Cayley tables? Textbooks often use the symmetries of polygons/solids to introduce group theory, however, I ...
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Who defined group representation in the modern way?

The modern definition of group representation is a homomorphism between a group $G$ and the group $GL(V, K)$ of some vector space over the field $K$. But as far as I know, when Frobenius developed ...
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Who discovered the topological proof of Nielsen-Schreier theorem?

The celebrated Nielsen-Schreier theorem in group theory says subgroup of a free group is free. This was proved for finitely generated subgroups of free groups by Jakob Nielsen in 1921, which involved ...
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Injection of Bernoulli numbers into topology

The Bernoulli numbers appear in the Harer-Zagier formula enumerating gluings of polygons, the Kervaire-Milnor formula for the order of homotopy groups for n-spheres, and (with the connection to the ...
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2 answers
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Did Evariste Galois create the entire group structure concept?

Did Evariste Galois create the entire group structure concept? If yes, were "super-sets" of groups (e.g. rings or vector spaces) created on top of Galois's work? When and by who? If no, did Galois ...
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Reflections in the 18th century

It is well known that the theory of reflections was considerably developed during the 19th century with the development of group theory (e.g. Klein) and the theory of transformations. However, I'm ...
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First appearance of modern definition of a group

What is the first appearance in print of the modern definition of an abstract group? To qualify, it should be a formal definition, contain the word "elements" (so Burnside's 1897 restriction to "...
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Books on Group Theory between 1885-1900

While reading the book of Burnside, the history gave interest to me to see further the old books on group theory. It will be a great pleasure if one can suggest some books on group theory published ...
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What came first? The kernel from vector spaces or from group theory?

In studying vector spaces we learn about linear transformations from one vector space to another and in particular the kernel of such a transformation. When learning about group theory we also learn ...
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3 votes
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209 views

History of the Wreath product

Why is the wreath product so named? If possible, please provide a citation.
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12 votes
1 answer
277 views

When was the modern field theory approach to Galois theory developed?

It is well known that Galois, and other mathematicians around that time, considered Galois groups to be permutation groups and approached Galois theory in this manner. At some point the theory took a ...
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22 votes
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How did group theory enter quantum mechanics?

How did the physicists in the 1920s become aware of the importance of group theory in quantum mechanics? Was group theory already part of the physics curriculum at that time, perhaps in connection to ...
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13 votes
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What group theoretic results were known for several special cases before the general definition of a group was established?

Many results in group theory were proven for permutation groups before the general definition of a group was established (for example: Lagrange's theorem, Sylow's theorems). However, permutation ...
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