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An important aspect of group theory is how groups can be represented as linear operators acting on vector spaces. While I understand how this works and how the (at least basic) tools are defined, what I struggle to understand is why this sort of thing is so useful.

To my understanding, working on a "group representation" is kind of like saying that the elements of the group have been "promoted" so that I can now "add" them together, "multiply them by scalars", and have them "act on other vectors". In other words, if before I could only write things such as $gh$ for $g,h\in G$, now I can write things such as $(\alpha g+h)v$ with $\alpha\in\mathbb F$ and $v\in V$.

What I find odd is that now results obtained using this additional structure are used to infer things about the original groups. For example, I read in this other answer how the classification of finite groups "would be unthinkable without representation theory", and in here and here about many other applications. I don't understand why this should be the case: why does adding "fake structure" (as in, structure that was not originally in the group under study) help classify groups, or help understanding groups in any other way?

To be clear, I'm not asking why are group actions an important part of group theory, like is done for example in this question. Rather, I ask why specifically actions on linear spaces reveal so useful. What is it about empowering a group with an additional abelian structure (plus scalars etc) that makes is so useful in providing insight about the group structure itself?

I am also not asking why is group representation theory important or useful for applications, or why groups are. Rather, I am asking why specifically adding a linear structure help us better understand the structure of groups.

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  • $\begingroup$ Linear representations are easy to work with (linear algebra is easier than group theory), easily decomposed (e.g., any representation of a finite group decomposes as the direct sum of (repeated copies of) a finite collection of irreps), and finite groups have enough of them to be interesting. There's also the idea (e.g., Peter-Weyl) that the continuous linear representations of a compact group act like the representations of a finite discrete group. You could look at more f.g. $G$-modules, but general modules are much more complicated than vector spaces. $\endgroup$ – anomaly Mar 9 at 14:33
  • $\begingroup$ @anomaly yes but please note that I am not in any way doubting the usefulness of representing groups in linear spaces, or of finite groups for their own sake. I know they are useful for many things. My question is why does adding abelian structure to a group help to characterize/classify it. It seems odd to me that to characterize something, I end up adding "fake structure" (structure that was not there in the first place) to it, and this nonetheless ends up telling me things about the original object $\endgroup$ – glS Mar 9 at 14:38
  • $\begingroup$ in other words, if my goal is, say, classify groups, I should be able to do so using only the structure of the groups themselves, right? So why does adding a linear structure that wasn't there in the first place end up being so useful for this? Why can I not classify the groups without it (or can I)? $\endgroup$ – glS Mar 9 at 14:41
  • $\begingroup$ For instance, maybe you want to solve the word problem in the given finitely presented group. While this is impossible in general, it is possible for subgroups of matrix groups. In short, matrix representations of finitely-generated groups allow you construct interesting finite quotients. There are many other applications, including that of infinite-dimensional unitary representations of finitely-generated groups. On the other hand, "classifying" groups is an impossibly hard task. $\endgroup$ – Moishe Kohan Mar 9 at 17:55
  • $\begingroup$ Centralizers in groups can be complicated objects. Centralizers in algebras are subalgebras. This, for a subgroup $G\subset\mathrm{GL}_d(K)$, the centralizer of $G$ is the intersection of $G$ with some subalgebra of $M_d(K)$. Corollary: chains of centralizers in $G$ are finite, and actually of length $\le d^2$. $\endgroup$ – YCor Mar 9 at 17:59
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This is a very good and fair question. I confess that for a long time I thought that there was no way you could gain any insight from a representation that you couldn’t gain from grappling with the group itself directly, simply because all you are doing is considering a homomorphic image of the group. It took a long time to get it through my head that they are actually useful.

The first step in recognizing that was to finally recognize the use of group actions, another instance in which my feeling was that you couldn’t possibly gain any new insight, given that a group action amounts to just an embedding into a symmetric group. Slowly but surely I came to understand that group actions give you a good way to understand a group because it allows you to think about the group as doing something, as opposed to as an abstract entity. You can use the way in which the group acts to understand the group. You can use it to identify subgroups as stabilizers, you can use it to recognize that something is not normal by how the action works, etc. It may take a while, but it turns out that it gives another way to look at the group, and of course the more ways you have to think of and look at the group, the better: you can play one against the other.

Once you recognize that group actions are a way to get into the group, then you turn to actions of a group not on a set, but on a structured set, and in particular actions that respect the structure. That gives you even more handles on the action. And as it happens, linear transformations are some of the best understood and most fruitful instances of “actions on a structured set”. So that trying to understand groups via their actions on vector spaces becomes something to try/want to have.

In addition to this, there is another important philosophy in algebra: you understand an algebraic object better by how it maps and acts on other things than just by staring at the object. So you will have more luck in understanding a ring by looking at its category of modules than just by staring at the ring; and you have more luck in understanding a group by considering homomorphic images of the group and group actions than just by staring at the group.

Once you combine those things, you end up with the idea that you want to add structural handles to the group because it gives you more “ways in”; it helps you see the group “in action” instead of as a static abstract object. And that the best kinds of actions you can look at are the ones that act on sets with lots of structure, and are best understood. (Felix Klein for example wanted to study groups as “systems of symmetries” of some objects/geometry/etc).

Once you land there, linear representations are an obvious thing to try. There is so much structure in the image that you can still transport back to the original group that you end up being able to gain a lot of information about the group by how it acts on vector spaces.

This approach proved useful early on; there are two famous results that were originally proven with representation theory that took a long time to prove without (and that turned out to require a lot of very strong machinery to prove without): I confess I can’t recall the second one, but the one I always remember is Burnside’s $p^{\alpha}q^{\beta}$ theorem, which was not proven without representation theory until after the Odd Order Theorem, and with many of those same ideas and techniques. Isaac’s recent book in Group Theory (not the one on character theory) can be seen as a book-long exposition of developing a lot of those tools with the objective of culminating with that representation-theory-free proof.

So, why does this “new ‘fake’ structure” help us understand the group? Because it gives us more ways to look at the group, more ways to think about the group elements, and a different way to conceptualize the group, not just as a set with an operation, but as an object that is acting on a structured object in well-understood ways.

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    $\begingroup$ The other result you were thinking of might be the existence of Frobenius kernels in Frobenius groups. $\endgroup$ – Tobias Kildetoft Mar 10 at 7:30
  • $\begingroup$ this is very interesting, thanks. There is still something that I don't quite get in all of this though. From what you are saying, it would seem that indeed, in theory, all results obtained via representation theory could have also been obtained directly by just "staring" at the original object. Now, I understand that in practice representation theory might help a lot in gaining insight, but in principle, it should be possible to avoid it altogether, is that correct? $\endgroup$ – glS Mar 10 at 21:05
  • $\begingroup$ Also, it would seem like the success of the choice of vector spaces here is more about how much is already known about vector spaces, than about the type of structure of vector spaces themselves. Would you agree with this statement? $\endgroup$ – glS Mar 10 at 21:05
  • $\begingroup$ @glS: I would agree partially with your last comment. Knowing a lot about the structure you are acting on helps substantially. But another important aspect is that if you have more structure, and you are respecting that structure, that tells you more about the group; the very fact that it “can” act while respecting a lot of structure in turn tells you that the group has to have a lot of structure. For vector spaces, we know a lot about what happens when you act while respecting the structure, but it is also a very good balance of “a lot of structure” and “not too much”. $\endgroup$ – Arturo Magidin Mar 10 at 21:15
  • $\begingroup$ @glS In principle, it would seem like you could get all the information from staring at the object. But if you look at the sheer amount of work that you need to go through to prove, for example, the $p^{\alpha}q^{\beta}$ theorem, it can quickly become apparent that not having access to those actions really curtails your ability to make general statements and prove them. What you would have a hard time proving just by “staring at an object” are general statements about objects that can act in certain ways. It allows you to consider all such groups, rather than stare at them individually. $\endgroup$ – Arturo Magidin Mar 10 at 21:18

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