Talk:History of group theory

Under construction

Just to let other editors know: the article is definitely a work in progress.

If you want to help before the end of September, it might be best to either discuss it here, or make sure your text is easy to fit into the provisional outline here. There is *no* need to correct typos or bad grammar. Virtually all of the "new" paragraphs will be rewritten. Supplying references is helpful, but marking unreferenced, wikify, peacock, etc. problems is not helpful as the rewrite will include toning down the language and including careful citations and wikilinks. It was just hard to summarize 130 years of one of the larger disciplines of mathematics in one shot!

Parts that I could use help with are often indicated in HTML comments. In particular, if anyone knows and cares about continuous groups, I am happy for the help, especially the more "analysis" style, especially 1960-2000.

Time outline

The rough outline is supposed to be:

• Pre-history
• 1850-1870
• 1870-1900
• 1900-1940 (1900-1920 + 1920-1940?)
• 1940-1960
• 1960-1980
• 1980-2000
• 2000-∞

Each time period is roughly broken up into finite, discrete, and continuous. In 1850-1870 there is basically only finite, 1870-1900 there is basically only finite and continuous with some very cautious investigations into "discontinuous groups". 1900-1940 is big because I am lazy. If it divides naturally into 1900-1920 and 1920-1940, I think that is fine. Most of the good stuff I can think of will divide that way, I just hadn't done it that way in my draft. Too much happens in 1940-1950 to allow 1940 to be too long, and too much happens in 1960-1980 to even fit in that one section.

Here are some reasons to divide at these marks:

• 1850 is roughly the birth of group theory
• 1870 is roughly the birth of Lie theory.
• 1900 is roughly the birth of character theory
• 1940 is roughly the advent of Brauer
• 1960 is roughly the advent of Feit-Thompson. It is followed Gorenstein's leadership, a slew of sporadics etc. so its very hard to stop until 1980
• 1980 is roughly the end of the first generation classification
• 2000 is a nice round number

I think 1940 is roughly the advent of Mackey too. 1930 is roughly the advent of Hall, but it sounded like an odd division (and some of his papers are late 20s).

Did anything happen in 1920? I don't have histories sorted by year, and nothing pops into mind, but it might look more even.

Topical outline

Finite can be divided into two: simple/character theory/permutation/linear and soluble/p-groups/formations. The former studies the complex internal structure of a group, and the latter studies the complex interactions of groups that are individually somewhat trivial. Currently wikipedia leans a little heavy on the former (probably because the CFSG is so high profile).

Discrete is most of the infinite groups which act on discrete things (like trees, sheets of a cover, lattices). Maybe divide it into:

• Combinatorial/Geometric group theory
• Infinite soluble groups (this pretty close to studying "group rings of infinite groups")
• Infinite abelian groups (also, but in a very different way!)
• Varieties of groups

Continuous is primarily the Lie theory, but should also handle a lot of the topological group stuff like:

• compact groups
• locally compact groups
• profinite groups
• algebraic groups

I think in my notes algebraic bounces between Discrete and Continuous. Basically algebraic groups act both on discrete and (positive characteristic-)continuous objects, so its a little shifty. Reflection groups are particularly bad since they are fundamental to both discrete and continuous groups. I think I list them as discrete.

Todo

• Obviously, rewrite everything 1870-2008
  • Fix tone, most of the language just lifted from sources, sometimes impartial things like publisher's book reviews!
  • Fix wikilinks, ensure all names are wikilinked first time, make a list of missing biographies
  • Fix sourcing, once the statements worth sourcing are determined, try to find uniform sources
  • Equal/due weight! Try harder to give each discipline its fair share.
• Split the 1700-1870 stuff more carefully. Most of this is well written, but some topics are given short schrift, and some incredibly important 1850-1870 events are omitted because they don't fit into the history of Galois theory masquerading as a history of group theory
• Find good historical sources for:
  • Groups in topology
  • Combinatorial group theory
  • Profinite groups
  • Post-Mackey continuous groups
• Treat Basse–Serre theory like the CFSG, "completed" in 1960-1980, "fruition" in 1980-2000

Ok, that's all for now. JackSchmidt (talk) 18:39, 4 September 2008 (UTC)

An excellent survery of the history of group theory in the early and mid 20th century is

• Chandler and Magnus, The history of the combinatorial group theory,

I'll be happy to fill in some holes in the coverage using the information contained therein. For early history of Lie theory we have a monograph

• Thomas Hawkins, The emergence of the theory of Lie groups

(I heavily used it in writing the history section at "Lie group"). Another excellent source on Lie groups and algebraic groups is

• Armand Borel, Essays in the history of Lie groups and algebraic groups.

A summary is in the "Historical note" to Bourbaki's "Lie groups and Lie algebras", Chapters 4–6. Arcfrk (talk) 21:11, 9 September 2008 (UTC)

Thanks! I'll check out Chandler-Magnus and Hawkins. I've been using Borel's book already, but I think it is perhaps a little eclectic (or at least not organized in the ideal way to lift an overview) so Hawkins should help organize my thoughts a little better. I think the C-M reference will be a huge help. All I know of combinatorial g.t. is just incidental from computational g.t. or from desperate attempts to understand the topological view of modular representation theory.

I'm planning to do another big push on Thursday. By the end of it, I hope to have solved most of the huge problems listed. Each field will be given its proper weight, but will probably still be incredibly sketchy and may show poor taste in which topics are highlighted.

After that, I'll definitely want your help editing, not just on the talk page. In particular, tastefully choosing the topics for each 20yr period for continuous groups and discrete groups is probably beyond me.

I want to get a believable (if poor taste) version up just so that everyone who edits the article can appreciate how huge the topic is, and how impossible it would be to give any individual event/movement its proper treatment. Instead, I hope we will be able to link to dedicated articles for each of these movements. CFSG and even Bass–Serre theory has its own page. I think a page on Mackey's contributions to continuous groups is being drafted. I definitely believe many more similar pages exist or could be written. Indeed, if I am forced to chop up the beautiful history of Galois theory, then I will make sure it has its own page where it can stand unmolested. JackSchmidt (talk) 21:42, 9 September 2008 (UTC)

C-M rules, thanks! This will make it much easier for me to put in a believable mock-up with real sources. I think it also cites a few references that could help at Group (mathematics) eventually too. I'll read it until Thursday, but that will be too soon for me to really absorb it, so I'll still need to depend on you to fix my second draft, especially with regards to the more subtle matters. Hopefully my choices will be much more tasteful as I will just lift them from C-M! Thanks again, the first section has been a great read already. JackSchmidt (talk) 01:28, 10 September 2008 (UTC)

Some early observations

Well done JackSchmidt on the drafting of all of this new material. Let me offer a few editorial comments from a lay perspective (I am not a mathematician and have not studied group theory but I have read a number of popular texts on the origins of group theory and the classification of simple finite groups). Thoughts as follows:

(1) Can you re-think the names of the various sub-headings? At the moment I see that you are referring to general periods of time such as "early 19th century" or "late 20th century" and even "later 20th century". I was wondering whether it is possible to be a little more descriptive than this in the headings to allow the reader to identify the achievements of the studied period more precisely? I realise that this is easier for me to request than for you to implement in practice as you are trying to draw together several strands of disparate mathematical development into a coherent whole. Maybe have a look at this article on a particular phase in Roman politico-legal history where you will see that the authors have tried to break the material out both thematically and chronologically. Closer to home there is this geomtery article which combines a mixture of approaches.

Absolutely. "Later" and "Late" were almost too embarrassing to save. I wanted to avoid explicitly listing years (since a publish date might be 1981, but it was announced in 1979, so being tied down to a specific year might be a hassle; rather it fit the this 20 year period's theme and wasn't too many years away from being in it).

I think the two examples you gave show how to do this very well. "Birth of group theory (c. 1840-1870)" "Birth of Lie theory (c. 1870-1900)" or so. The names might be hard to decide on, but the format looks very good. It will also force me to figure out the 1900-1940 name or split to 1900-1920, 1920-1940. JackSchmidt (talk) 13:43, 5 September 2008 (UTC)

(2) Can you space the photographs of the various notable personages out a little bit more as they seem to bulk up the start of the article? I appreciate that most of these are pre-20th century but maybe there is a way of re-arranging more aesthetically?

Part of the problem is the photographs were added before my huge block of text. I'll add the photos explicitly to the todo list. Probably a combination of adding more to the later sections, and possibly moving one or two photos down. JackSchmidt (talk) 13:43, 5 September 2008 (UTC)

(3) Is it worth adding more detail on the long project to classify simple finite groups? Also, should you add in some more detail on the work of Gorenstein and Conway? What about mathematical moonshine - is it worth making a mention? A few weeks ago I added in a reference in the bibliography to the new Marcus du Sautoy book which I recently read and which (somewhat diffusely) contains quite detailed historiography on various aspects of the development of group theory. You may find this text useful although you may feel that it places undue emphasis on recent European experience.

Well certainly more detail than is currently here! It might still be less detail than you want. I am hoping to use "summary style" here whenever possible, so that all interesting developments are mentioned, most are briefly described, and most are wikilinked to *full* articles giving much more in-depth histories. I'll probably completely the sketchy outline before fleshing out most of the topics. Also, once the sketchy outline is in place, it will be much easier for any editor to improve a particular paragraph (I will attempt to do my rewrites paragraph by paragraph). 13:43, 5 September 2008 (UTC)

(4) Try to eliminate judgmental words such as "excitement" and "remarkable".

Absolutely. The language was usually taken from the sources, and meant to give an overview of what the reader of the *new* (not yet written) text should get out of it. Instead of saying it is remarkable, I want to actually mention true, sourced, objective facts which the reader should find remarkable without being told so. For now though, I was just trying to lay down some sort of outline of what one might say. Part of the problem is deciding what to include: both remembering/finding the good stuff, and removing the less important stuff that only I find interesting. Since I wasn't sure if I'd end up deleting everything I wrote, I didn't spend much time "polishing" what I wrote (indeed, some of it is not even in sentence form!). JackSchmidt (talk) 13:43, 5 September 2008 (UTC)

Overall, this is looking very good. Kind regards--Calabraxthis ^(talk) 06:17, 5 September 2008 (UTC)

Thanks very much for looking it over. The heading help especially should give it a more professional look immediately. JackSchmidt (talk) 13:43, 5 September 2008 (UTC)

Errors: Early 19th century: Development of permutation groups

(1) At present (3 April 2019) the section "Early 19th century: Development of permutation groups" contains the statement:

"Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni."

This statement is incorrect. Ruffini's pamphlet of 1801 does not contain the phrase l'assieme delle permutazioni. (Here is a link to Ruffini's 1801 pamphlet: Ruffini, Paolo (1802). Della soluzione delle equazioni algebraiche determinate particolari di grado superiore al quarto [On the solution of certain determinate algebraic equations of degree higher than four] (in Italian). Modena, (Italy): Società Tipografica.)

This is partially due to the fact that assieme is not a noun, but it can be used both as an adverb where it means "together", and as a preposition where it means "together with". (See: Wiktionary: assieme.) The correct word is insieme, which means "set" and "group" (see: Wiktionary: insieme); this word does appear in several places in Ruffini's 1801 pamphlet. However, the phrase l'insieme delle permutazioni does not appear in Ruffini's 1801 pamphlet.

(2) At present (16 April 2019) the section "Early 19th century: Development of permutation groups" contains the statement:

"His first publication on group theory was made at the age of eighteen (1829), … ."

This statement is incorrect. In 1829, Galois did submit, for publication, two papers on the resolution of equations. However, the reviewers decided that Galois' submissions lacked sufficient detail and told him to revise and resubmit them. So Galois apparently withdrew the two papers. See: Neumann, Peter M. (2011). The Mathematical Writings of Évariste Galois. Zürich, Switzerland: European Mathematical Society. p. 2.

However, in 1829, one paper by Galois was published. It was a paper about continued fractions: Galois, Evariste (April 1829). "Démonstration d'un théorème sur les fractions continues périodiques". Annales de Mathématiques pures et appliquées (in French). 19: 294–302.

VexorAbVikipædia (talk) 18:46, 16 April 2019 (UTC)