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 $$K=\{1\}\cup\big\{G\setminus\bigcup_{g\in G}gHg^{-1}\big\}.$$ 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.

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.

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 $$K=\{1\}\cup\big\{G\setminus\bigcup_{g\in G}gHg^{-1}\big\}.$$ 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 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.

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 $$K=\{1\}\cup\big\{G\setminus\bigcup_{g\in G}gHg^{-1}\big\}.$$ 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.