Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Tarski monster group
Résumé
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.
Let be a fixed prime number. An infinite group is called a Tarski monster group for if every nontrivial subgroup (i.e. every subgroup other than 1 and G itself) has elements.
is necessarily finitely generated. In fact it is generated by every two non-commuting elements.
is simple. If and is any subgroup distinct from the subgroup would have elements.
The construction of Olshanskii shows in fact that there are continuum-many non-isomorphic Tarski Monster groups for each prime .
Tarski monster groups are an example of non-amenable groups not containing a free subgroup.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.