\titleofpresentation{Quasi-permutation representations of  finite groups}
\presenter{Houshang Behravesh}

If $G$ is a finite linear group of degree $n$, that is,
a finite group of automorphisms of an $n$-dimensional complex
vector space (or, equivalently , a finite group of non-singular
matrices of order $n$ with complex coefficients), we shall say
that $G$ is a quasi-permutation group if the trace of every
element of $G$ is a non-negative rational integer. The reason for
this terminology is that, if $G$ is a permutation group of degree
$n$, its elements, considered as acting on the elements of a
basis of an $n$-dimensional complex vector space $V$, induce
automorphisms of $V$ forming a group isomorphic to $G$. The trace
of the automorphism corresponding to an element $x$ of $G$ is
equal to the number of letters left fixed by $x$, and so is a
non-negative integer. Thus, a permutation group of degree $n$ has
a representation as a quasi-permutation group of degree $n$. See
\cite{behravesh8}.

By a quasi-permutation matrix we mean a square matrix over the complex
field $\mathbb C$ with non-negative integral trace. Thus every permutation
matrix over $\mathbb C$ is a quasi-permutation matrix. For a given finite
group $G$, let $p(G)$ denote the minimal degree of a faithful permutation
representation of $G$ (or of a faithful representation of $G$ by
permutation matrices), let $q(G)$ denote the minimal degree of a faithful
representation of $G$ by quasi-permutation matrices over the rational field
$\mathbb Q$, and let $c(G)$ be the minimal degree of a faithful representation
of $G$ by complex quasi-permutation matrices. See \cite{behravesh1}.

By a rational valued character we mean a character $\chi$ corresponding to
a complex representation of $G$ such that $\chi(g) \in {\mathbb Q}$ for all
$g \in G$. As the values of the character of a complex representation are
algebraic numbers, a rational valued character is in fact integer valued.
A quasi-permutation representation of $G$ is then simply a complex
representation of $G$ whose character values are rational and non-negative.
The module of such a representation will be called a quasi-permutation
module. We will call a homomorphism from $G$ to $GL(n,{\mathbb Q})$ a
rational representation of $G$ and its corresponding character will be
called a rational character of $G$.
%Let $r(G)$ denote the minimal
%degree of a faithful rational valued character of $G$.
It is easy to see that
%$r(G) \leq c(G)$ and
\[c(G) \leq q(G) \leq p(G) \]
where $G$ is a finite group.

We will give algorithms for calculating $p(G)$, $q(G)$ and $c(G)$
where $G$ is a finite group and also when $G$ has a unique
minimal normal subgroup. Also we will give  the old and  new
results on the quasi-permutation representation of finite groups.
Finally we will state  a question from Brian Hartley.


\begin{ourbiblio}{9}
\bibitem{behravesh1}
{J. M. Burns, B. Goldsmith, B. Hartley, R. Sandling}. On quasi-permutation
representations of finite groups. {\rm Glasgow Math.J. 36 (1994), 301-308}.

\bibitem{behravesh8}
{W. J. Wong}. Linear groups analogous to permutation groups.
{\rm J.Austral. Math.Soc. Ser.A 3 (1963), 180-184}.
\end{ourbiblio}

\presenteraddress{Department of Mathematics,
University of Urmia,
Urmia, Iran}

\email{hbehravesh@yahoo.com}
