\title{ Bounding the size of permutation groups and complex linear groups of odd order}
\author{Geoffrey R. Robinson,\\
Institute of Mathematics,\\
University of Aberdeen}
\date{August 2010}
\maketitle


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\noindent {\bf Introduction:} Numerous results exist in the literature bounding the orders
of finite odd order subgroups of permutation groups and complex linear groups (see, for example,
(Gow, \cite{gow}), where such bounds were used to bound the number of characters
in a $p$-block of a finite group). Our aim here is to point
out that these bounds can be substantially improved if we place restrictions on the smallest
prime divisor of the group order.

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Given an odd prime $p,$ we define constants $\alpha(p)$ and $\beta(p)$ as follows:

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\noindent i) $\alpha(3) = \sqrt{3}, \alpha(5) = 5^{\frac{1}{4}}$ and $\alpha(p) = [p(2p+1)]^{\frac{1}{2p}}$ for $p \geq 7.$\\

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\noindent ii) $\beta(p) = [p(2p+1)^{2}]^{\frac{1}{2p}}$ if $p \equiv 2$ (mod $3$)
and $\beta(p) = [p(2p-1)^{2}]^{\frac{1}{2p-2}}$ otherwise.


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We will prove:

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\noindent {\bf Theorem:} \emph{ a) Let $G$ be a subgroup of odd order of the symmetric group $S_{n}.$
Then $|G| \leq \alpha(p)^{n-1},$ where $p$ is the smallest prime divisor of $|G|.$\\
\noindent b) Let $G$ be a finite subgroup of odd order of ${\rm GL}(n,\mathbb{C})$  and let $p$ be the smallest prime divisor
of $|G|.$ Then:\\
\noindent i) $[G:F(G)] \leq \alpha(p)^{n-1}$ and \\
\noindent ii) $G$ has an Abelian normal subgroup $A$
with $[G:A] \leq \beta(p)^{n-1}.$\\}