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Version 1 |
| \PMlinkescapeword{lattice} |
\PMlinkescapeword{lattice} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
| \PMlinkescapeword{satisfies} |
\PMlinkescapeword{satisfies} |
| \PMlinkescapeword{subgroup} |
\PMlinkescapeword{subgroup} |
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| A \emph{Tarski group} is an infinite group $G$ |
A \emph{Tarski group} is an infinite group $G$ |
| such that every non-trivial proper subgroup of $G$ is of prime order. |
such that every non-trivial proper subgroup of $G$ is of prime order. |
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| Tarski groups are also called \emph{Tarski monsters}, |
Tarski groups are also called \emph{Tarski monsters}, |
| especially in the case when |
especially in the case when |
| all the proper non-trivial subgroups are of the same order |
all the proper non-trivial subgroups are of the same order |
| (that is, when the Tarski group is |
(that is, when the Tarski group is |
| a \PMlinkname{$p$-group}{PGroup4} for some prime $p$). |
a \PMlinkname{$p$-group}{PGroup4} for some prime $p$). |
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Ol'shanski\u{\i}\cite{ol1,ol2} showed that Tarski groups exist,
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Ol'shanskii\cite{ol1,ol2} showed that Tarski groups exist,
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| and that there is a Tarski $p$-group for every prime $p > 10^{75}$. |
and that there is a Tarski $p$-group for every prime $p > 10^{75}$. |
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| From the definition one can easily deduce |
From the definition one can easily deduce |
| a number of properties of Tarski groups. |
a number of properties of Tarski groups. |
| For example, |
For example, |
| every Tarski group is a simple group, |
every Tarski group is a simple group, |
| it satisfies the minimal condition and the maximal condition, |
it satisfies the minimal condition and the maximal condition, |
|
it can be generated by just two elements,
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it can be generated by just $2$ elements,
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| its subgroup \PMlinkname{lattice}{Lattice} is \PMlinkname{modular}{ModularLattice}, |
its subgroup \PMlinkname{lattice}{Lattice} is \PMlinkname{modular}{ModularLattice}, |
| and it is periodic but not locally finite. |
and it is periodic but not locally finite. |
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|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{ol1} |
\bibitem{ol1} |
| A.\ Yu.\ Ol'shanski\u{\i}, |
A.\ Yu.\ Ol'shanski\u{\i}, |
| {\it An infinite group with subgroups of prime orders}, |
{\it An infinite group with subgroups of prime orders}, |
| Inv.\ Akad.\ Nauk SSSR Ser.\ Mat.\ 44 (1980), 309--321. |
Inv.\ Akad.\ Nauk SSSR Ser.\ Mat.\ 44 (1980), 309--321. |
| (Russian) |
(Russian) |
| \bibitem{ol2} |
\bibitem{ol2} |
| A.\ Yu.\ Ol'shanski\u{\i}, |
A.\ Yu.\ Ol'shanski\u{\i}, |
| {\it Groups of bounded period with subgroups of prime order}, |
{\it Groups of bounded period with subgroups of prime order}, |
| Algebra i Logica 21 (1982), 553--618. |
Algebra i Logica 21 (1982), 553--618. |
| (Russian) |
(Russian) |
| \end{thebibliography} |
\end{thebibliography} |
