Authors: 

M.~Hertweck and W.~Kimmerle

Universit\"at Stuttgart, Pfaffenwaldring 57, 
D-70550 Stuttgart, Germany 

Fax: (49 711)685-5304

Title:

Coleman automorphisms of finite groups

Abstract:

An automorphism $\sigma$ of a finite group $G$ whose 
restriction to any Sylow subgroup equals the restriction of some inner 
automorphism of $G$ shall be called {\em Coleman automorphism}, named for
D.~B.~Coleman, who's important observation from \cite{col:64} especially shows
that such automorphisms occur naturally in the study of
the normalizer $\mathcal{N}$ of $G$ in the units $\mathcal{U}$ 
of the integral group $\ZZ G$. Let $\outcol{G}$ be the image of these
automorphisms in $\out{G}$. We prove that
$\outcol{G}$ is always an abelian group (based on previous work of E.~C.~Dade,
who showed that $\outcol{G}$ is always nilpotent).

We prove that if no composition factor of $G$ has order $p$ (a fixed prime), 
then $\outcol{G}$ is a $p^{\prime}$-group. If $\opn{p}{G}=1$, it suffices 
to assume that no chief factor of $G$ has order $p$.

If $G$ is solvable and no chief factor of $G/\opn{2}{G}$ has order $2$, then
$\mathcal{N}=G\mathcal{Z}$, where $\mathcal{Z}$ is the center of
$\mathcal{U}$. This improves an earlier result of S.~Jackowski and 
Z.~Marciniak. 

Current status: submitted for publication