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Character theory is shockingly effective at proving things in group theory - even statements that can be entirely stated in the language of groups. Below are two famous examples:

Burnside's Theorem: Let $G$ be a finite group whose order has exactly two prime factors, i.e. $|G|=p^aq^b$ for distinct primes $p,q$ and positive integers $a,b$. Then $G$ is solvable.

Frobenius's Theorem: A finite group $G$ is called a Frobenius group if there is a non-trivial subgroup $H$ of $G$ (known as the Frobenius complement of $G$) such that if $g\notin H$, then $H \cap gHg^{-1} = \{1\}$. In this case, the Frobenius kernel of $G$, $K$, is defined as $$K=\{1\}\cup\bigl(G\setminus\bigcup_{g\in G}gHg^{-1}\bigr).$$ Frobenius's Theorem states that if $G$ is a Frobenius group with $H$ and $K$ as above, then $K$ is a normal subgroup of $G$, hence $G=K\rtimes H$.

The former has a relatively simple proof via character theory, but also a decidedly more difficult proof that only uses group theory.

The latter, meanwhile, has no known character-free proofs, although again the existing proofs are not too hard.

Informally, I feel like math has some sort of "conservation of difficulty": A proof may look simple, but if you unravel enough of the logical dependencies, a significant theorem should have "hard" theorems or lemmata underlying it.

The group-theoretic proof of Burnside's Theorem is hard enough to convince me that any proof of it should have a lot of non-trivial math behind it, perhaps hidden under the rug.

But… where is it hiding in the character-theoretic proof? The best I can spot is that there are a lot of algebro-number-theoretic properties underlying characters of finite groups (over $\mathbb{C}$). Does that mean the "difficulty" is ultimately coming from the glorious properties of the complex numbers? And trying to prove it without complex characters means you lose all of those tools?

P.S. As with many things, I realized that Terry Tao has touched on this idea of "conservation of difficulty", albeit in the context of solutions to PDEs.

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    $\begingroup$ There was a lot of effort put into finding an "elementary" proof of the prime number theorem (one avoiding the use of complex numbers), but then when one was found by Selberg/Erdős, not much came of it. It could just be that the character theoretic proof of Frobenius's theorem is the "right" one. $\endgroup$
    – Sam Hopkins
    Commented Sep 25 at 22:13
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    $\begingroup$ In the proof of the classification of finite simple groups, it's just as remarkable how character theoretic methods (including modular character theoretic methods such as Glauberman's proof of the $Z^*$ theorem) give out long before the main structural methods take hold. In the end, I think the best approach is to use whatever methods you can lay your hands on for a first proof, and then try to see what's really necessary or best later on. $\endgroup$
    – Dave Benson
    Commented Sep 25 at 22:34
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    $\begingroup$ @WillSawin even such simple and common thing as independent non-uniform Bernoulli ("join every two vertices by an edge with probability $n^{-\theta}$") is not about sizes but about weighted sums, which look involved without probabilistic view and natural with it. Reducing more advanced tools like entropy, or Lovasz local lemma, to counting the sizes of sets seems practically impossible for me. $\endgroup$
    – Fedor Petrov
    Commented Sep 26 at 1:59
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    $\begingroup$ @FedorPetrov I'm not talking about the probabilistic view but rather the probabilistic foundation. The point I was trying, perhaps badly, to make is that the idea of thinking probabilistically, of considering what proportion of objects (maybe with some weighting) satisfy a property when asked whether an object satisfying the property exists, is what's crucial, and the foundations of probability theory (e.g. measure theory) are usually irrelevant. The point is that the difficulty is not hiding in the foundations (or in the proof of the Lovasz local lemma) but in finding the right idea. $\endgroup$
    – Will Sawin
    Commented Sep 26 at 10:48
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    $\begingroup$ Note that Burnside's Theorem is an easy consequence of the following: A finite non-abelian simple group does not have a conjugacy class of primepower length $> 1$. In fact the latter is what the character theoretic proof of Burnside's theorem shows. Nevertheless, as far as I know, this latter fact does not have a character-free proof yet. $\endgroup$
    – Peter Mueller
    Commented Sep 26 at 13:09

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This doesn't exactly directly answer the question, but I think a lot of the magic of representation theory is in this principle. It's sort of a theme of 20th century and modern math that understanding the entire category of, say, $G$-modules simplifies the understanding of $G$ itself, and it's not a uniquely rep-theoretic phenomenon: as another for-example in algebraic geometry, understanding the category of schemes "over $X$" is somehow easier and more powerful than directly trying to understand $X$. Maybe one can say that the difficulty hides in setting up all this abstract machinery in the first place? Characters of $G$-irreps are certainly much more abstract an idea than $G$ itself, and take a good deal of infrastructure to start using.

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