How to find groups with given properties

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How to find groups with given properties

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[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. There is another article with [[How to find continuous groups with given properties|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.

[note article incomplete] 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.[/note]

==== 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. [[Basic examples of groups|Here is a list of well-known groups]], with brief discussions of the kinds of properties they can be expected to have. [[Properties of groups|And here is a list of some important group properties]], including several examples of which groups have each possible set of properties.

Another approach is to rely on group libraries in a computer algebra package such as GAP. It is often possible to define the property in question in GAP notation and quickly scan large sets of groups to see if any of them has this property.

There is also a more general page about [[Useful examples and counterexamples|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 <add text="more concretely"> as the group of symmetries of a regular n-gon</add>.) For more details, see [[Presentations of groups|this article about presentations of groups]].

==== Method 3: Take interesting subgroups of existing groups. ====

Some groups are conveniently described as [[Interesting subgroups of less interesting groups|subgroups of larger groups]].

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

There are [[New groups from old|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 of groups|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 [[Groups of symmetries|a space of its own]].

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