Revision of How to build groups from Mon, 09/03/2009 - 12:08

Quick description

This page gives a summary of several techniques that are commonly used for answering questions of the form, ``Does there exist a group $G$ such that ...?'' The emphasis is on discrete groups. See here for a discussion of topological groups and Lie groups.

Prerequisites

A basic knowledge of group theory, such as would be taught in typical first course on the subject.

Disclaimer

This page is far from complete, and its subpages are even more so. It is written by a non-expert and would undoubtedly benefit from some expert attention.

Method 1: Check the desired conditions against a library of well-known examples

The most obvious way of determining whether a group exists with a certain property is to look at the groups you know and see whether it has the property in question. This page contains a list of well-known groups, with brief discussions of the kinds of properties they can be expected to have. There is also a more general page about off-the-shelf examples in mathematics.

Method 2: Write down a presentation.

An important way of specifying a group is by means of generators and relations. For instance, the dihedral group of order $2n$ is the group with two generators $a$ and $b$ subject to the relations $a^n=b^2=1$ (where $1$ is the identity of the group) and $bab=a^{-1}$ . (It can also be defined as the group of symmetries of a regular n-gon.) More details about this method can be found here.

Method 3: Take interesting subgroups of existing groups.

A few examples of groups that are conveniently described as subgroups of larger groups can be found here.

Method 4: Build new groups out of old ones using products and quotients.

There are several different kinds of products, described here. As well as descriptions of the products themselves are descriptions of the kinds of uses to which they are typically put. Quotients are described here.

Method 5: Take the group of symmetries of some other mathematical object.

This method could be regarded as a special case of Method 1, but it is sufficiently important to deserve a space of its own.