21
votes
3answers
689 views

Who named “Quotient groups”?

Who decided to call quotient groups quotient groups, and why did they choose that name? A lot of identities such as $$\frac{G/A}{B/A}\cong \frac{G}{B}$$ suggest that whoever invented the notation ...
7
votes
1answer
88 views

Who first proved the fundamental theorem of finitely generated (or finite) abelian groups?

The fundamental theorem of finitely generated abelian groups (or maybe just finite abelian groups) is well-known and can be found in just about any text on the theory of groups or abstract algebra. ...
2
votes
1answer
64 views

How did the Symmetric group and Alternating group come to be named as such?

The Dihedral group makes sense, "Di" means two, and "hedral" means.. shape I think (I've just realised how much of what I think words mean are guesses based on experience) like a "polygon" is a 2d ...
1
vote
0answers
39 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
2
votes
1answer
54 views

Necessary and sufficient conditions for the embeddability of a semigroup in a group

According to wikipedia, The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in (Malcev 1939).[5] Though theoretically important, the ...
3
votes
0answers
90 views

Classification of Groups

Arthur Cayley classified all groups of order $4$ and $6$ in 1854, and groups of order $8$ in 1858. What about groups of order $2,3,5,7$?. These are prime numbers, and the most basic theorem in group ...
1
vote
1answer
83 views

Why are regular $p$-groups called “regular?”

In the concept of regular $p$-Groups, what does "regularity" stand for? What is "regular" in such groups? I would like to know idea behind defining these groups, and naming these groups "regular." ...
6
votes
0answers
189 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...
3
votes
2answers
113 views

Who did first use the term “simple” in group sense and why?

Who did first use the term "simple" in group sense? More appreciated if I learn the original word(might be in French, Galois???)... and also... Why do you think that this term is chosen ...
3
votes
1answer
99 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
181 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 ...
27
votes
3answers
855 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 ...
53
votes
1answer
2k 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
193 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
238 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 ...
9
votes
3answers
775 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 ...
47
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 ...