4
$\begingroup$

For a fixed prime $p$, the Sylow $p$-subgroups of a given finite group are all conjugate. Here are some more examples of situations in which we find that subgroups of a finite group defined by a certain property are all conjugate:

  1. The Hall $\pi$-subgroups of a finite solvable group, where $\pi$ is a fixed set of primes.
  2. The nilpotent Hall $\pi$-subgroups of a finite group, where $\pi$ is a fixed set of primes.
  3. The core-free maximal subgroups of a finite solvable group
  4. The complements of a normal Hall subgroup of a finite group (Schur-Zassenhaus)
  5. The Carter subgroups of a finite group
  6. The nilpotent injectors of a finite solvable group

If we look at the way these conjugacy theorems are proved, we might be inclined to think that they're all in some sense "derived" from the Sylow conjugacy theorem, but I can't help but wonder if there might be some alternate interpretation of such phenomena.

It's possible to define the terms "core-free", "maximal" and "solvable" without referring to the primes that divide $|G|$. And yet if we look at a proof of the third fact in the list above, we'd see that it uses Sylow's theorems.

Question: Is there an alternate interpretation of this widespread phenomena, and possibly an "arithmetic-free" or "Sylow-free" way to prove some of these theorems?

Dream: Is there a "purely group-theoretic" characterization of Sylow subgroups? i.e. "$H\subseteq G$ is a Sylow $p$-subgroup for some prime $p$ if and only if it satisfies group-theoretic property P."

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Is "purely group-theoretic" defined, or is it the sort of thing that you just know when you see it? For example, I guess you don't want any references to order, even though I think of order as a group-theoretic property. Of course it cannot be purely intrinsic to $H$, since the same $p$-group can be Sylow in one group and not in another; but what is one allowed to use about the interaction of $H$ with $G$? $\endgroup$
    – LSpice
    Commented May 3, 2023 at 12:35
  • $\begingroup$ No, "purely group-theoretic" isn't well-defined. But an example of what I mean is: 1) A finite group is solvable if and only if its derived subgroup terminates. 2) A finite group is solvable if and only if it has a $p$-complement for every prime $p$. The former is "purely group-theoretic", whereas the latter is "arithmetic". Of course, we need to use the fact that the groups we are considering is finite if we want to talk about Sylow subgroups. But I want something that doesn't make explicit reference to the prime factorization of the order of the group. $\endgroup$
    – semisimpleton
    Commented May 3, 2023 at 12:39
  • $\begingroup$ (cont.) I'm fine with using combinatorial arguments like "If $X$ is a finite set, then an injection $f:X\rightarrow X$ is also a bijection", or like the so-called "pigeonhole principle", etc. $\endgroup$
    – semisimpleton
    Commented May 3, 2023 at 12:42
  • $\begingroup$ You might like to look at the theory of formations. $\endgroup$
    – Geoff Robinson
    Commented May 14, 2023 at 9:05

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .