TITLE: Automorphism group of a tournament is solvable
QUESTION [5 upvotes]: $(a)$ Let $X$ be a tournament, i.e. $X$ is a directed complete graph. Denote $V(X)$ the vertex set of $X$. An automorphism of $X$ is a bijection $V(X) \to V(X)$ preserving orientation. Prove that the automorphism group $\text{Aut}(X)$ is solvable.
$(b)$ Prove, in fact, that the following statements are equivalent:
Every group of odd order is solvable. $\iff$ $\text{Aut}(X)$ is solvable for every tournament $X$.

I am having trouble getting started, in particular how could I compute the automorphism group of a given tournament? This seems very hard if the number of vertices is large.

REPLY [4 votes]: This question has already be answered by Sameer - the argument is essentially as follows:

If $X$ is a tournament, we show that the group $G=\operatorname{Aut}(X)$ has odd order.  Otherwise, if it has even order then it must contain an element of order $2$, by Cauchy's theorem.  But such an element must necessarily transpose two vertices, and hence reverse the arrow between them, meaning that it is not an automorphism of $X$.  This is a contradiction, so $G$ must have odd order.
Now, by the Feit-Thompson Theorem, we conclude that $G$ is solvable.

This is a valid proof, but the Feit-Thompson theorem has a 200-page proof, and seems overpowered for this problem?  Is there a simpler proof that avoids the Feit-Thompson Theorem?
The answer is - no there isn't!  In fact, this problem is equivalent to the Feit-Thompson theorem thanks to a result of J. W. Moon stating that every group $G$ of odd order is the automorphism group of some tournament.  Thus, if there were a simpler proof of this statement, it would immediately give a much simpler proof of the Feit-Thompson theorem:

Let $G$ be a group of odd order. By Moon's result (which is much shorter to prove than FT), $G$ is the automorphism group of some tournament.
By our putative simple proof, $G$ must be soluble.

Since no short proof of the Feit-Thompson Theorem is known, there is also no known short proof of this result that avoids the use of the Feit-Thompson Theorem.
If you found a short proof that the automorphism group of any tournament is soluble, it would be big news!