Tagged Questions
3
votes
1answer
72 views
Historical definition of a group
Wikipedia states that van Dyck (1882) was the first to give the definition of a group in the modern way. Before this, what were some of the original axioms or conditions for groups? I mean, how were ...
3
votes
0answers
146 views
Which theorem did Poincaré prove?
Two related elementary facts in group theory are sometimes called Poincaré's theorems.
If $H\lneq G$ and $[G:H]<\infty$, then there is $N\leq H$, $N\lhd G$ such that $[G:N]<\infty$.
The ...
24
votes
3answers
501 views
Where does the word “torsion” in algebra come from?
Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
46
votes
1answer
1k views
How was the Monster's existence originally suspected?
I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence.
For ...
4
votes
0answers
125 views
Clarification: intersection of a finite number of subgroups of finite index and Poincaré
From Scott's book Group Theory
$1.7.10.$ (Poincaré) The intersection of a finite number of subgroups of finite index is of finite index.
My question is: Did Poincaré prove the Theorem as stated ...
7
votes
3answers
185 views
The Hopfian property for groups
Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context ...
6
votes
3answers
385 views
Where is the name “coset” in group theory from?
One of the most important application of "coset", I think, is to prove the Lagrange's theorem, which was not originally stated in the group theoretic terms. In some textbooks I have read about ...
38
votes
6answers
2k views
What kind of “symmetry” is the symmetric group about?
There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections ...
