When mathematicians study specific classes of groups how do they chose which realization of them to use?

Question

By cayleys theorem every group $G$ is isomorphic to a permutation group over the elements of $G$ yet it seems that in many instances trying to express certain groups in this way only makes it more complicated to study them. But at the same time after reading articles on specific terminology for permutation groups e.g. https://en.wikipedia.org/wiki/Block_(permutation_group_theory) it seems people have developed specific techniques for dealing with permutation groups in particular, which leads me to believe there are some instances in which a permutation group is the most useful way to represent a group. How does one know when this is?

Answer 1

This question is quite open to opinion, but I had fun thinking about it, so I'll give my answer:

It is a slightly strange question, because a lot of work is put into finding different ways to represent a group. This is because different ways of representing a group can be used to see different information about the group.

There are some properties of a group that actually depend on the many representations of a group. A character table for example carries information about all the linear representations of a group.

The only time you really have to settle on a representation is when using a computer, anything you can do by hand you can probably do quite easily in any representation you choose. But thinking about how you'd like to represent a group can be interesting.

Typically the choices are:

• permutation group (a subgroup of $Sym(\Omega)$ for some set $\Omega$)
• matrix group (a subgroup of a $GL(n,\mathbb{F})$ for some field $\mathbb{F})$
• projective matrix group (a subgroup of a $PGL(n,\mathbb{F})$ for some field $\mathbb{F})$
• group presentation

There are definitely others, but I believe these are the most common and well studied.

I'll list a few factors here that help make the decision and might add to the list as I think of them:

• Personal preference
• How the group given - changing representation is often more work than it's worth
• What you actually want to do - sometimes algorithms working on a given representation are too slow or practically impossible. For example, if you ask a computer to return the size of a finite group then it will have a far easier time with a permutation group than with a group presentation.
• How you want to present results - typically people seem to like to see an easy to write group presentation more than a list of permutations that generate the group.
• How easy it is use a given representation - a permutation group for example is easier to compute with if it has small degree or a small base.
• what the representation offers - different representations offer different useful language. Character theory is an incredibly useful field of study that comes from linear representations, while permutation groups offer properties like transitivity and primitivity.