\titleofpresentation{On Conjugacy Classes of Finite Groups with a regard towards Groups of Squarefree Order}

\presenter{Anna Torstensson}  
Conjugacy classes are a vital tool in the analysis of groups. A
condensed piece of information about a group is its class number, that
is its number of conjugacy classes. Starting with small numbers
mathematicians have tried to catalogise all groups having a given
class number. In my talk I will give brief history of this problem. The starting point was when Landau in 1903
managed to show that there is only a finite number of
non-isomorphic finite groups with class number $k$, and also derived an
upper bound on the group size as a function of $k$. In \cite{torstenssonPoland}
Poland extends the complete list of groups with class number
$k$ from $k \leq 5$ to $k \leq 7$. Using computerised methods this was
later pushed forward to cover $k \leq 9$ (references can be found in
\cite{torstenssonKomissarcik}) and considering only simple groups further to $k
\leq 12$ by Komissarcik in \cite{torstenssonKomissarcik}. The reason to limit the
search to simple groups was that for $5 \leq k \leq 9$
the largest groups with a given class number are all simple. Hence one
might guess that this is true also for larger $k$ even though this
seems to be unknown.

There has also been successful attempts to attack
other types of groups. For example $p$-groups where it is known that a
group of order $p^m$ has $(n+r(p-1))(p^2-1)+p^e$ conjugacy classes
where $r$ is some non-negative integer and $m=2n+e$, $e= 0$ or $1$ depending on the parity
of $m$. This result can be found in \cite{torstenssonPoland2}.

For the rest of the talk we
will go in the opposite direction and instead of $p$-groups consider
groups of squarefree order. We will look at an explicit formula for the
class numbers groups of a given squarefree order can have. As a
by product of the analysis we will also see that a group of squarefree
order having $k$ conjugacy classes can have no more than $k^3$ elements. There are bounds on the group order in the general case, but
in contrast to this bound they are far from sharp. All these results are proved in detail in \cite{torstensson}.

It is of course a substantial confinement to consider only groups of
squarefree orders. It should be noted however that most group orders
are covered, as $\lim_{n \to \infty} \frac{Q(n)}{n} =
\frac{6}{\pi^2}$, where $Q(n)$ denotes the number of squarefree
numbers $\leq n$. (See \cite{torstenssonHardyWright}.) In addition the techniques
developed here can hopefully be modified in order to cover other
types of groups which have relatively simple presentations.

\begin{ourbiblio}{99}  

\bibitem{torstensson} A. ~Torstensson, On Conjugacy Classes of Groups of Squarefree Order, preprint Lund University 2006. Submitted for journal publication.

 \bibitem{torstenssonHardyWright}  
G. H.~Hardy and E. M.~Wright, 
The Number of Squarefree Numbers in {\it An introduction to the theory
  of Numbers} 4th (corrected) ed. Clarendon Press (Oxford) 1962.
%
\bibitem{torstenssonKomissarcik} 
E.A.~Komissarcik, 
Simple Groups with fewer than thirteen conjugacy classes, 
{\it Soviet Math. Dokl.\/} {\bf 18}(6) (1977), 1536--1539.
%

\bibitem{torstenssonMurtyMurty} 
M.R.~Murty and V.K.~Murty, 
On Groups of Squarefree Order, 
{\it Math. Ann.\/} {\bf 267} (1984), 299--309. 
%
\bibitem{torstenssonNewman} 
M.~Newman, 
A bound for the number of conjugacy classes in a group, 
{\it J. London Math. Soc.\/} {\bf 43} (1968), 108--110.
%
\bibitem{torstenssonPoland}  
J.~Poland, 
Finite Groups with a given number of Conjugate Classes,
{\it Canad. J. Math.~\/} {\bf 20} (1968), 456--463.
%
\bibitem{torstenssonPoland2}  
J.~Poland, 
Two problems on Finite Groups with $k$ Conjugate Classes,
{\it J. Austral. Math. Soc.~\/} {\bf 8} (1968), 49--55.
%
% \bibitem{yourname2} Another paper;

% \bibitem{yourname3} Etc.

\end{ourbiblio}

\presenteraddress{The Royal Institute of Technology, Stockholm, Sweden}  

\email{annator@kth.se}

\web{www.math.kth.se/~annator/}  


