There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results actually have relatively easier proofs to the ones I will outline below. More specifically, I am interested in whether these results may be proven without the aid of the powerful subnormality theory.
The first, and perhaps most beautiful result, is due to Horosevskii:
Let $\sigma\in Aut(G)$ where Aut($G$) denotes the automorphism group of a non-trivial finite group $G$. Then the order of $\sigma$ in Aut($G$) in less than $\left|G\right|$.
The second result, though perhaps more specialized, actually is an important ingredient in the purely group-theoretic proof of Burnside's $p^aq^b$-theorem:
Let $t$ be an involution in a finite group $G$, and assume that $t\not\in {\bf{O}}_2(G)$, where ${\bf{O}}_2(G)$ denotes the 2-core of $G$. Then there exists an element $x\in G$ of odd prime order such that $txt=x^{-1}$.
(I think that this result is due to Matsuyama but please do not take my word for this because I am not completely certain. For definiteness, I will refer to this result as the "result on involutions".)
Now the only proof I know of the second result (on involutions) relies on Baer's theorem in subnormality theory. For reference, Baer's theorem states that:
Let $H$ be a subgroup of a finite group $G$. Then $H\subseteq {\bf{F}}(G)$ if and only if $\left\langle H,H^x \right\rangle$ is nilpotent for all $x\in G$.
(Here, ${\bf{F}}(G)$ denotes the Fitting subgroup of $G$ and $H^x$ denotes the conjugate of $H$ by $x$; that is, $H^x = \{x^{-1}hx|h\in H\}$.)
The proof (at least the one I know) of the result on involutions uses Baer's theorem, but actually, it really only uses a very special case of Baer's theorem: the subgroup $H$ in the statement of Baer's theorem is chosen to be $\{1,t\}$ in the proof, where $t$ is the involution quoted in the result. (The proof also uses a very easy fact about dihedral groups.)
This leads me to wonder whether there exists a "Baer-free" proof of the result on involutions. More specifically, my question is:
- Question 1: Does there exist a proof of the result on involutions independent of subnormality theory?
It would be interesting if such a proof existed since Baer's theorem relies on the non-trivial Wielandt "zipper lemma" (which I will quote at the end of my question) - and the zipper lemma really looks unrelated to the result on involutions.
Now Horosevskii's theorem relies on another non-trivial consequence of subnormality theory: Lucchini's theorem. For reference, Lucchini's theorem states that:
Let $A$ be a cyclic proper subgroup of a finite group $G$, and let $K=\mbox{core}_G(A)$. Then $\left|A:K\right|<\left|G:A\right|$, and in particular, if $\left|A\right|\geq \left|G:A\right|$, then $K>1$.
What is amazing, at least to me, is that a fact that one might consider fundamental about automorphisms (Horosevskii's theorem) relies on a result (Lucchini's theorem) that at least on a first glance seems very specialized. My second question is nearly identical to my first:
- Question 2: Does there exist a proof of Horosevskii's theorem independent of subnormality theory?
(What follows is not really subsumed in my question, but it may be useful for answering my question.)
Succinctly, the proof of Baer's theorem (at least the proof I know) relies on the amazing Wielandt "zipper lemma" which states:
Suppose that $S\subseteq G$, where $G$ is a finite group, and assume that $S$ is subnormal in $H$ for every proper subgroup $H$ of $G$ that contains $S$. If $S$ is not subnormal in $G$, then there is a unique maximal subgroup of $G$ that conains $S$.
Likewise, the proof of Lucchini's theorem relies on Zenkov's theorem on intersections of abelian subgroups, the proof of which in turn relies on Baer's theorem. Zenkov's theorem states:
Let $A$ and $B$ be abelian subgroups of a finite group $G$ and let $M$ be a minimal member of the set $\{A\cap B^g|g\in G\}$. Then $M\subseteq {\bf{F}}(G)$.
(Note that "$B^g$" denotes the conjugate of $B$ by $g$; that is, $B^g=\{g^{-1}bg|b\in B\}$.)
(Note: I am well aware that there may be other theorems due to Baer, Lucchini and Horosevskii that are referred to as "Baer's theorem", "Lucchini's theorem" or "Horosevskii's theorem". However, I hope that by stating the results above, no confusion arises. For proofs of the theorems quoted, please see the book Finite Group Theory by I. Martin Isaacs. More specifically, please see Theorem 2.9., Chapter 2, Section 2A, page 50 for the statement and proof of the Wielandt "Zipper Lemma", Theorem 2.12., Chapter 2, Section 2B, page 55 for the statement and proof of Baer's theorem, Theorem 2.18., Chapter 2, Section 2D, page 61 for the statement and proof of Zenkov's theorem, and Theorem 2.20., Chapter 2, Section 2D, page 63 for the statement and proof of Lucchini's Theorem.)
Thanks!
