a repository of mathematical know-how

Group theory front page

Quick description

This article gives an introductory description of group theory and links to Tricki articles about various aspects of groups.

Click here to go straight to the links

Note iconIncomplete This article is incomplete. There's still lots more to be done. We need to decide on the overall structure of the area.

General discussion

The definition of a group makes precise the intuitive idea of the set of symmetries of an object. By performing two symmetries in series one gets a third symmetry, and any symmetry can be undone by means of another symmetry. Taking these ideas in the abstract gives the axioms for a group: a set with an associative binary operation, with an identity element and inverses. These are some of the minimal assumptions that one usually makes when doing algebra—the most familiar binary operations, multiplication and addition, satisfy these properties (as long as one works in the right number system)—and so one can think of groups as some of the most basic algebraic objects. Indeed, we usually refer to the binary operation on a group as multiplication (with an important exception listed later); when we multiply group elements g,h together, we call the result gh; and the identity element of an identity group is sometimes denoted 1.

To recapture the notion of symmetry, one defines an action. To say that a group G acts on a set X is to identify every g\in G with a map  X\to X. Of course, the action should respect group multiplication, and so it seems natural to require that f_{gh}=f_g\circ f_h. In fact, this definition has the odd consequence that gh corresponds to applying the symmetry h first and then g afterwards. If you don't like this, then alternatively you can define your action to have the property that f_{gh}=f_h\circ f_g. The first definition is called a left action and the second is a right action. The difference may seem arbitrary, but it is important not to mix left actions and right actions up!

The fact that one has to distinguish between left actions and right actions comes from the fact that the order in which we multiply group elements matters: group multiplication is not required to be commutative. In this respect, group multiplication differs from the usual addition and multiplication of numbers, but some of the simplest examples of symmetries that we can think of are not commutative.

Example 1

Symmetries of a triangle

If the operation on a group G does happen to be commutative then G is called Abelian. Abelian groups are particularly important and well-understood. In the case of abelian groups it is traditional to use additive notation, rather than multiplicative, so the group operation is denoted by + and the identity element is denoted 0.

The group of symmetries of the triangle is finite, as we saw above, and indeed the group of symmetries of any finite object will also be finite. Finite groups are much-studied, and the classification of finite simple groups—the "building blocks" of finite groups—is one of the most important achievements in the history of mathematics.

So far, we have taken the definition of a group to be purely algebraic. However, it is often more natural to think of a group as being equipped with a topology, particularly when trying to make sense of uncountable groups. Two of the most important classes of topological groups are Lie groups, which have the topology of a manifold, and profinite groups, whose topology is totally disconnected. One can recover the purely algebraic notion of a group by imposing the discrete topology. Infinite discrete groups are often studied using the techniques of combinatorial or geometric group theory.

As mentioned above, one is often interested in a group G because of its action on an object X. If X is just a set then the action of G is said to be by permutations. Another very important case is when X is a vector space, in which case an action of G by linear transformations is called a representation of G. This is the study of representation theory.

Articles about group theory

  • Elementary arguments in group theory

    • To prove facts about finite groups, use 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.)

    • Normal subgroups

      • 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. )

    • Subgroups of finite index

      • 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. )

  • Group actions

    • 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. )

    • Use topology to study your group

  • Examples of groups

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

  • Constructing groups with particular properties

    • 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. )

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


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.

Post new comment

(Note: commenting is not possible on this snapshot.)

Before posting from this form, please consider whether it would be more appropriate to make an inline comment using the Turn commenting on link near the bottom of the window. (Simply click the link, move the cursor over the article, and click on the piece of text on which you want to comment.)