8
votes
2answers
453 views

Origin of the Socle of a module

Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?
5
votes
2answers
414 views

When did the meaning of the term “metabelian” change?

I just realised that the meaning of the term "metabelian", when applied to groups, or Lie algebras, seems to have changed over years. (These days, it means that $[[G,G],[G,G]]$ is trivial, while in ...
11
votes
3answers
696 views

The role of the Automatic Groups in the history of Geometric Group Theory

What is the role of the theory of Automatic Groups in the history of Geometric Group Theory? Motivation: When I read through the "Word Processing in Groups" I was amazed by the supreme beauty and ...
27
votes
4answers
2k views

What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...
43
votes
3answers
2k views

Groups that do not exist

In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and ...
16
votes
1answer
1k views

On a theorem of Galois

I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois : Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...
8
votes
1answer
428 views

History of a conjecture/problem non-inner automorphisms of order p in finite p-groups

The following conjecture/problem posed in The Kourovka Notebook in 1973 by Ya. G. Berkovich: Problem 4.13. Prove that every finite non-abelian $p$-group admits an automorphism of order $p$ which is ...
18
votes
4answers
2k views

How did “Ore's Conjecture” become a conjecture?

The narrow question here concerns the history of one development in group theory, but the broader context involves the sometimes loose use of the term "conjecture". This goes back to older work of ...
5
votes
2answers
335 views

Hall's treatment of algebraic operations

Marshall Hall, in his famous book Theory of Groups, does not always require a binary operation be "well-defined", i.e. an operation is a relation instead of a function (there might be more than one ...
4
votes
1answer
305 views

Residual finiteness of fundamental groups of surfaces.

What is a simple way to prove that for any compact two-dimensional surface $S$ and an element $g$ in $\mathbb \pi_1(S)$ there exists a finite index normal subgroup $\Gamma\subset \pi_1(S)$ such that ...
18
votes
2answers
1k views

Does the amenability problem for Thompson's group $F$ predate 1980?

The first place where the amenability problem for Thompson's group $F$ appears in the literature is, I believe, 1980 in a problems article by Ross Geoghegan. I have heard, however, vague comments to ...
13
votes
1answer
743 views

A synopsis of Adyan’s solution to the general Burnside problem?

Where can I find a high-level overview of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent? Additionally: If possible, would an expert ...
15
votes
2answers
2k views

Galois theory timeline

A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective ...
28
votes
2answers
2k views

Why are parabolic subgroups called “parabolic subgroups”?

Over the years, I have heard two different proposed answers to this question. It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a ...