Is every permutation group isomorphic to a 'familiar' group?

Question

This may be a very simple question, but I do not know how to approach it.

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G (in other words, isomorphic to a permutation group, a permutation group just being a subgroup of a symmetric group).

What I am curious about is:

Is every permutation group isomorphic to some 'familiar' group?

(Feel free to read as "finite permutation group" if it makes a difference.)

What I mean by a 'familiar' group is a group that isn't typically viewed as a group of permutations, such as cyclic groups, dihedral groups, groups of matrices like the general linear group, etc. Phrased another way, is there a 'nice' description of every permutation group in terms of common groups and not depending on a presentation with generators and relations or reference to the object that the permutation groups acts on?

It seems there should be some hope for this, since a fundamental notion of groups is that they describe symmetries. But perhaps this does not mean that every permutation group is a 'nice enough' or 'interesting enough' permutation that it receives a nice name to go by.

I really hope my question is clear, even if the language may not be the best with which to describe what I'm after.

What I have so far is that if our permutation group is abelian, it seems it should be easy to classify it by the fundamental theorem of finitely generated abelian groups, thus providing an answer in that case.

In addition, some subgroups of symmetric groups are recognized as familiar groups, as shown here.

Thanks in advance for the input.

EDIT: What I want to know, if this is any better (and maybe I should have stated it this way in the first place), is just if there are any permutation groups for which the sole, or by far best, description is as a group of permutations, instead of in terms of any kind of combination of groups that do have a description not in terms of the set of permutations on some object.

Answer 1

It's pretty hard to answer this question when 'familiar' isn't defined further.

Every finite group (and thus every permutation group) has a composition series, which is unique in the sense that the length and composition factors of any two composition series are the same up to permutation and isomorphism. If you define familiar groups to be simple groups, then these quotients are the familiar pieces you're looking for.

Defining familiar as simple is a stretch, though. You'd be hard pressed to find someone who found the Held group especially recognizable.

The other problem is that two nonisomorphic groups can have the same composition factors. The factors do help you break the group into smaller, more recongizable chunks, but they don't tell you how those chunks interact.

There are so many variations on combinations of smaller groups that it is difficult to imagine how we could recognize them all without group presentations. Another way of breaking down a group into recognizable groups is by looking at its Sylow subgroups and seeing how they interact (this is local group theory). But for this, we need to understand $p$-groups enough to call them recognizable. My answer to this question should give you an idea of the magnitude of what we're dealing with just in the world of $p$-groups.

Take for example $\text{SmallGroup}(16,3)$. There is no other name for that group, as far as I know. It is isomorphic to $(\mathbb{Z}_4\times\mathbb{Z}_2)\rtimes \mathbb{Z}_2$, which shows us it can be split into those recognizable pieces. This decomposition uses semidirect products, however, which is basically the same as the relations / presentation concept we are trying to avoid.

So in summary, there are many a lot of different ways to divide a group up into pieces that are as familiar as possible, to gain understanding about it and how it works. However, to achieve a full description of the group, most of the time we need to use specific relations in a group presentation to show the way those pieces fit together.

Answer 2

Just looking at finite simple groups, the classification is fairly complex, and I doubt you'd call all of the examples, "familiar."

In particular, the odd examples are the so-called Sporadic Groups. These are a $26$ specific finite groups that do not fit into broad infinite classes. Indeed, some of them were initially discovered during the attempt to classify the finite simple groups - they were not initially found in terms of the groups of symmetries of any mathematical object.

One of the stranger results is that almost all groups of order less than 2000 have order 1024. There are $49,487,365,422$ distinct groups of order $1024$. That is more than 99% of the distinct groups of order less than 2000. Certainly, some of these must not be "familiar."