\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Melody Chan}
%
%
\medskip
\noindent
%
%
{\bf The Maximum Distinguishing Number of a Group}
%
%
\vskip 5mm
\noindent
%
%
%
%
Let $G$ be a group acting faithfully on a set $X$. The distinguishing
number of the action of $G$ on $X$, denoted $D_G(X)$, is the smallest
number of colors such that there exists a coloring of $X$ where no
nontrivial group element induces a color-preserving permutation of
$X$. In this paper, we show that if $G$ is nilpotent of class $c$ or
supersolvable of length $c$ then $G$ always acts with distinguishing
number at most $c+1$. We obtain that all metacyclic groups act with
distinguishing number at most 3; these include all groups of
squarefree order. We also prove that the distinguishing number of the
action of the general linear group $GL_n(K)$ over a field $K$ on the
vector space $K^n$ is 2 if $K$ has at least $n+1$ elements.
\bye