Tarski group
TarskiGroup
2006-03-14 10:01:12
2006-03-22 12:36:49
Definition
Tarski monster
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A \emph{Tarski group} is an infinite group $G$
such that every non-trivial proper subgroup of $G$ is of prime order.
Tarski groups are also called \emph{Tarski monsters},
especially in the case when
all the proper non-trivial subgroups are of the same order
(that is, when the Tarski group is
a \PMlinkname{$p$-group}{PGroup4} for some prime $p$).
Alexander Ol'shanskii\cite{ol1,ol2} showed that Tarski groups exist,
and that there is a Tarski $p$-group for every prime $p > 10^{75}$.
From the definition one can easily deduce
a number of properties of Tarski groups.
For example,
every Tarski group is a simple group,
it satisfies the minimal condition and the maximal condition,
it can be generated by just two elements,
it is periodic but not locally finite,
and its \PMlinkname{subgroup lattice}{LatticeOfSubgroups} is \PMlinkname{modular}{ModularLattice}.
\begin{thebibliography}{9}
\bibitem{ol1}
A.\ Yu.\ Olshanskii,
{\it An infinite group with subgroups of prime orders},
Math.\ USSR Izv.\ 16 (1981), 279--289;
translation of Izvestia Akad.\ Nauk SSSR Ser.\ Matem.\ 44 (1980), 309--321.
\bibitem{ol2}
A.\ Yu.\ Olshanskii,
{\it Groups of bounded period with subgroups of prime order},
Algebra and Logic 21 (1983), 369--418;
translation of Algebra i Logika 21 (1982), 553--618.
\end{thebibliography}