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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...