Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $p$. There are many proofs of Sylow's existence theorem, but I wonder if it is possible to prove it via representation theory.
Here is a speculative proof strategy:
- Find a suitable $FG$-module $V$ (for some well-chosen field $F$), such that $\text{dim}_F(V)=|G|_{p'}$.
($|G|_{p'}$ is the $p'$-part of $|G|$, by which I mean the greatest divisor of $|G|$ that is not divisible by $p$. If $G$ had a Sylow $p$-subgroup (we know it does, but let's pretend we don't, since this is what we're trying to prove), then $|G|_{p'}$ would be the index of the Sylow $p$-subgroup.)
- Since $G$ acts on $V$, we can construct a crossed homomorphism $G\rightarrow V$. Prove that the kernel of this crossed homomorphism is a Sylow $p$-subgroup of $G$.
I imagine carrying out this proof strategy would involve modular representation theory. I don't know what $F$ should be, nor do I know how to construct a suitable $V$. We have to construct $V$ without assuming that $G$ has a Sylow $p$-subgroup. Does someone know how to (if it is possible at all) implement this proof strategy?
Edit: Since we already know that Sylow $p$-subgroups exist, we know that to implement this strategy, it would suffice to somehow construct something isomorphic or $G$-conjugate to the permutation module $F[G/P]$ (where $P$ is a Sylow $p$-subgroup of $G$), without assuming the existence of Sylow subgroups. Does anyone know how to do this?
