Groupprops:Introduction for group theorists

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This is a first-person article written by: Vipul
First-person articles give opinions of their authors, as long as these opinions are broadly endorsed by the wiki

Dear group theorists!

A very warm welcome to Groupprops. The Groupprops wiki would not have been possible without all the dedicated research that has gone on in group theory for more than hundred years, by people like you. Whether or not you have heard of this wiki, your contributions and work is a source of inspiration to the wiki.

As you navigate this wiki, you will find a number of interesting things here, and possibly also some things you may consider incorrect, inappropriate, or lopsided. The wiki is in its very initial stages and your feedback at this stage is very valuable. You can provide feedback on the feedback page or send it to me by email through the site (you need to create an account and sign in for both). If you don't want to create an account, you can email me at or

Below I have given a brief introduction to the concept of a wiki, and how Groupprops is as a wiki.

Wikis and groupprops

The wiki model and why it helps

The mathematical community, particularly the group theory community, has been well-served by a large number of well-written books, many research publications, many great teachers, and it remains active through activities like conferences, symposia, and many others. Many of these bring an induplicable contribution to the field of group theory. Groupprops does not in any way aim to replace or supplant these well-tested resources. We are trying to provide another type of value that complements the existing value.

I have thought long and hard about what genuinely new value a wiki can provide, and the organizational methodology of the wiki is testimony to the many experiments. Take a look, for instance, at the page pronormal subgroup or at the page characteristic implies normal. You'll see that the tools and kind of value that I try to provide in these pages is significantly different in nature from the value a textbook or lecture may provide. A textbook would define these terms in a broader flow of things, in a systematic manner. A lecture is more interactive. But there is a crucial element of value in the wiki pages that's not there in textbooks and lectures, and it is that element that we're working on.

What is that element? One part of it is: each term, each fact, has exactly one page to it. So if you're looking for a proof of the statement that every characteristic subgroup is normal, you'll get a full page's worth for it. Which means that you don't have to look in the corners or margins or scan through exercise lists or comments to get a brief mention of the fact. Everything sits on its own. Every term, every fact.

Built on this is the concept of strong internal linking. It's not feasible in a linear, textbook-like model, to have one page devoted to every term, one page devoted to every fact, and so on, because there isn't any natural way to order things, and referring back and forth is a pain. Textbooks often omit proofs by giving references to other books which have the proof, up to the page number and proposition number. So you fish out the other book, to the relevant page, read the proof, then get back. But with wikis, it's really simple. A little markup, often done automatically, allows one to link to a huge number of pages from a given page. This means easy and quick exploration. No more pulling out tomes from shelves to check a one-line definition.

With one thing, one page and strong internal linking comes the next feature: relate to everything else that's related. What really excites me about this wiki is that, despite having created almost all of it myself, I still learn a lot by surfing through it because navigating around using the many interrelationships and connections is so easy. Right now, the wiki is too small to provide the kind of ideas and inputs needed for frontier reesarch in group theory. But I'm confident that the approach and ideas behind the organization are going to make it a pretty useful tool in the future.

Some differences with other wikis

Further information: We're different from Wikipedia Mathematics is the most accurate of the sciences, and group theorists take great care to be accurate in stating things. So you may not be comfortable with unreviewed, raw content floating around on this wiki. For many of us, wikis have become synonymous with Wikipedia or similar projects, where loads of people come and edit and wrong facts often stay on for years.

Groupprops has been built on a wiki primarily because of strong internal linking and the other features a wiki model provides. Strong collaboration, i.e. having a number of people work together on a single article, hasn't happened on this wiki yet. But even when it does, the collaboration will be between people who love and appreciate group theory. This means that policy is designed keeping in mind people who have come to learn, share, or do research.

We do have the problem of content not being properly checked or reviewed, so errors may creep in. However, most of the statements can be checked by hand, and as experienced group theorists, I am sure you'll have no problem figuring out the wrong statements (and please correct errors if you find them!). By putting separate proof pages for everything, I'm trying to minimize the errors that occur because I simply assume something is true without writing down the proof.

Some odd things

Terminology local to the wiki

I've used a lot of terminology on the wiki that's local to the wiki-- it isn't standard in the group theory community and may never have been used formally in any research publication. The reason is simple: a lot of the organization of the wiki depends on principles and paradigms that require names and putting a name to things makes it easier to classify, relate and organize. Most of the terminology local to the wiki seems to be very aptly named, and may have been used informally already.

For instance, a transitive subgroup property is a subgroup property such that if H satisfies the property as a subgroup of K and K satisfies the property as a subgroup of G, H satisfies the property as a subgroup of G. Seems a reasonable name, and in fact, you might even guess the definition just from the statement that normality is not transitive.

My personal opinion is that the mathematics community has adopted too little terminology for the simple ideas and this may have led to a greater profusion of terminology at the level of higher ideas. This wiki doesn't follow the general economy principle of not giving a name to something that can easily be described. We've tried to give names to anything that occurs often enough. Sometimes, the name isn't really a name, it's a phrase. For instance maximal among normal Abelian subgroups is a self-evident name.

With the wiki, it's particularly easy to handle new terminology: just click on it and see what it means. No need any more to turn back to the introduction page or search for the reference book from which the terminology was taken.

As an experienced group theorist, you may have some knowledge about whether the wiki-local terminology has been used in a standard publication, or whether there is another term for it used in any standard publication. If you do, please let me know!

Lopsided development

Groupprops has a lot of articles in some areas, and very few in others. We have too many definitions, too few facts, and very few survey articles and expository articles. We lack references for a lot of definitions and facts, where references are needed or desired.

This lopsided development is partly because rather than try to shove in every aspect of group theory on the wiki, I have been concentrating in the last few months on working out better ways of organizing and navigating the wiki. Once this task is done to a reasonable extent, I shall begin in earnest to document a lot of stuff related to the ideas behind the classification of finite simple groups. I'll also be focussing on documenting more about particular groups, I'll work on linear representation theory and geometric group theory, and I'll put in more survey articles.

Your views on further directions of development, or areas where the wiki is weak or strong, are strongly desired.

Getting around this wiki

You probably want to look around and see what kind of content is available on the wiki. Here are some of the tools you could use:

  • search the wiki: You could use the search bar in the left column -- it's always available for use. Learn more about searching at Help:Searching Groupprops.
  • Lookup: In the left column, you see a long list of things under the lookup header. Clicking on any of these will take you to the category of all things of that type. For instance, if you're looking for definitions, click the Terms/definitions link on the left, and you go the Category:Terminology. Navigate the subcategories of this as you want.