Questions tagged [group-theory]
The study of algebraic structures known as groups.
26
questions
2
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0answers
161 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 ...
3
votes
1answer
123 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, ...
7
votes
2answers
605 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
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 ...
7
votes
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 ...
2
votes
0answers
139 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
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 ...
2
votes
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 ...
13
votes
0answers
311 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/...
9
votes
0answers
341 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” ...
0
votes
0answers
152 views
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 ...
7
votes
1answer
307 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
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 ...
3
votes
1answer
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 ...
3
votes
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 ...
6
votes
1answer
314 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
301 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
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 ...
5
votes
0answers
135 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
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 "...
5
votes
2answers
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 ...
11
votes
2answers
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 ...
3
votes
0answers
138 views
History of the Wreath product
Why is the wreath product so named?
If possible, please provide a citation.
12
votes
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 ...
21
votes
1answer
1k 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 ...
13
votes
2answers
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 ...
