Questions tagged [group-theory]

The study of algebraic structures known as groups.

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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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1answer
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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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1answer
115 views

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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1answer
170 views

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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1answer
263 views

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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1answer
191 views

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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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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How did Lie groups work their way into the Standard Model of Particle Physics [duplicate]

In a question answered on this site concerning how group theory worked its way into quantum mechanics it was proposed that this was done by Weyl while investigating the many electron problem. Yes, but ...
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1answer
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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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885 views

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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75 views

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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1answer
115 views

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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1answer
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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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1answer
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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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815 views

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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3answers
199 views

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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140 views

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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412 views

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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138 views

History of the Wreath product

Why is the wreath product so named? If possible, please provide a citation.
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1answer
191 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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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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175 views

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 ...