• Source: Tarski monster group

In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group G, such that every proper subgroup H of G, other than the identity subgroup, is a cyclic group of order a fixed prime number p. A Tarski monster group is necessarily simple. It was shown by Alexander Yu. Olshanskii in 1979 that Tarski groups exist, and that there is a Tarski p-group for every prime p > 1075. They are a source of counterexamples to conjectures in group theory, most importantly to Burnside's problem and the von Neumann conjecture.




Definition


Let



p


{\displaystyle p}

be a fixed prime number. An infinite group



G


{\displaystyle G}

is called a Tarski monster group for



p


{\displaystyle p}

if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has



p


{\displaystyle p}

elements.




Properties






G


{\displaystyle G}

is necessarily finitely generated. In fact it is generated by every two non-commuting elements.




G


{\displaystyle G}

is simple. If



N
⊴
G


{\displaystyle N\trianglelefteq G}

and



U
≤
G


{\displaystyle U\leq G}

is any subgroup distinct from



N


{\displaystyle N}

the subgroup



N
U


{\displaystyle NU}

would have




p

2




{\displaystyle p^{2}}

elements.
The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime



p
>

10

75




{\displaystyle p>10^{75}}

.
Tarski monster groups are examples of non-amenable groups not containing any free subgroups.




References


A. Yu. Olshanskii, 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.
A. Yu. Olshanskii, 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.
Ol'shanskiĭ, A. Yu. (1991), Geometry of defining relations in groups, Mathematics and its Applications (Soviet Series), vol. 70, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-1394-6

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• Daftar topik teori grup
• Grup-p
• Tarski monster group
• Alfred Tarski
• P-group
• List of things named after Alfred Tarski
• Simple group
• List of group theory topics
• Linear group
• Tarski's problem
• Torsion group
• Locally finite group