Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent:
- The set K of elements of G of order relatively prime to p (the p′-elements) form a subgroup
- If A and Ag are both subsets of P, then there is some x in CG(A) such that xg is in P
That the first implies the second is a silly trick: $(a^{-1} a^g)^{-1} = g^{-1} g^a \in P \cap K = 1$ for any $a \in P$ and $g \in K$ such that $a^g \in P$.
The second implies the first is not too hard (Frobenius normal p-complement theorem), but I'm trying to use this as a first example, and so don't want to have any prerequisites outside a very gentle undergraduate group theory course. Most of the rest of the talk is just using Sylow's theorem.
Is there a very low-tech, short, few-preliminaries proof that "absolutely no fusion" implies a normal p-complement?
I would be ok with assuming P is abelian, so that we get:
- If A and Ag are both subsets of P, then g in CG(A).
I'm also fine with assuming p = 2 so that "relatively prime to p" shortens to "odd".
I don't think using the transfer is appropriate, as it won't be used again, and the whole point of this proof is to motivate learning something else.
For a "no" answer to my question, it would be sufficient to convince me that transfer is needed (and nice if you can suggest a special case where it wouldn't be needed, other than P of order 2).