Questions tagged [group-theory]
The study of algebraic structures known as groups.
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questions
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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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1answer
151 views
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&...
3
votes
1answer
109 views
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 ...
2
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1answer
232 views
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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0answers
91 views
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 ...
3
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0answers
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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'?
3
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0answers
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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 ...
0
votes
1answer
91 views
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 ...
2
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0answers
93 views
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 ...
2
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0answers
64 views
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 ...
3
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0answers
179 views
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 ...
4
votes
1answer
183 views
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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2answers
720 views
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 ...
2
votes
1answer
164 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 ...
8
votes
1answer
303 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 ...
2
votes
0answers
356 views
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 $b$, ...
2
votes
1answer
347 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 ...
2
votes
1answer
364 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 ...
14
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0answers
415 views
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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0answers
369 views
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” ...
8
votes
1answer
497 views
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, ...
5
votes
4answers
1k 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 ...
3
votes
1answer
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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 ...
3
votes
1answer
137 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 ...
6
votes
1answer
419 views
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 ...
5
votes
1answer
364 views
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 ...
11
votes
2answers
985 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 ...
5
votes
0answers
187 views
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 ...
11
votes
3answers
312 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 "...
5
votes
2answers
169 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 ...
11
votes
2answers
528 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 ...
3
votes
0answers
169 views
History of the Wreath product
Why is the wreath product so named?
If possible, please provide a citation.
12
votes
1answer
247 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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1answer
2k views
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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2answers
207 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 ...
