Tricki
a repository of mathematical know-how

Revision of Group theory front page from Sun, 19/04/2009 - 19:10

Quick description

This article contains links to pages about techniques that are specific to group theory.

Note iconIncomplete This article is incomplete. This article could do with some general discussion about different kinds of proofs in group theory, and more links to subsidiary pages, whether or not those pages have been written yet. Also, when there are more articles in group theory, they will need to be categorized further and this page will have to be reorganized.
  • How to find groups with given properties Quick description. ( If you are trying to prove a statement about all groups with a certain property, then it helps to have a repertoire of examples and methods of construction, so that you can get a feel for whether the statement is true. This article is about various methods for building groups. )

    • Basic examples of groups Quick description. ( This article lists various important groups and families of groups, and discusses some of their properties. )

    • Presentations of groups Quick description. ( This article is about defining groups by means of generators and relations. )

  • How to use group actions Quick description. ( Many problems in group theory that do not explicitly mention group actions can be solved by means of a clever choice of action. This article gives links to articles on different types of group action and how they can be used to prove theorems. )

  • First pretend that a normal subgroup is trivial Quick description. ( When dealing with a group theory problem involving a normal subgroup H of a group G, pretend first that H is trivial, and solve this simpler problem. To then handle the general case, quotient everything by H and apply what you have just done. In many cases, this solves the problem "modulo H"; to finish the job, one has to figure out how to deal with some residual objects lying in H.)

  • A way of getting proper normal subgroups of small index Quick description. ( Given a group, it can be useful to know that there is a proper normal subgroup of small index, where 'small' is defined in terms of some finite invariant of the group. To obtain such a result, it suffices to show that the group has a non-trivial action on a sufficiently small finite set. These may arise from the internal structure of the group. )

  • Improving a subgroup of finite index by intersecting Quick description. ( Given a group G and a subgroup H of finite index, you can find a new subgroup of finite index with better properties by taking the intersection of H with other subgroups of finite index. This gives a method of constructing finite-index subgroups that are normal or characteristic in G. )

  • How to prove facts about finite groups by induction on the order Quick description ( One approach to proving results about finite groups is by using induction on the order of the group. The general idea is that one has many constructions of subgroups, such as the formation of the centre, or of centralizers of elements, or of normalizers of subgroups, to which one can hope to apply an inductive argument.)

Comments

Inline comments

The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.

Do we want these subarticles?

This page seems to me to be shaping up nicely, but I'd like to raise a question about all the dead links in these three paragraphs. The implication seems to be that there will be pages called "Abelian groups front page", "Finite groups front page", "Lie groups front page", "Profinite groups front page", "Combinatorial group theory front page" and "Geometric group theory front page". Is that the idea?

If it is (and it seems sensible), then at some stage I would probably want to rewrite this section slightly, in order to avoid the impression it currently gives that the Tricki focuses on subject matter (things like the definition of an Abelian group) rather than proof techniques. So I'd probably make those dead links into Wikipedia links and then have a paragraph saying something like "These special classes of groups have their own particular techniques, which are discussed and linked to from the following pages."

I suppose you're saying that

I suppose you're saying that you'd rather have pages with names like "How to solve a problem about finite groups", right? I'm all for that. I'll make the change at some point.

Yes. My first choice is

Yes. My first choice is titles like "To prove that a small group is simple, look at the sizes of the conjugacy classes," but that won't work for more general pages that discuss or link to several techniques. So then titles such as you suggest (perhaps I'd go for "problems" rather than "a problem" but that's a very minor quibble!) would come into play.

I updated the link to the How

I updated the link to the How to prove facts about finite groups by induction on the order page to take into account the new name of that page, and moved it so as to be under the How to solve problems about finite groups link.