This article is about the three unusual infinite groups F, T and V found by Thompson. For the sporadic simple group, see Thompson sporadic group.
In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.
The Thompson groups, and F in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.
It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved
von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.
Higman (1974) introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.
term Thompsongroup or Thompson'sgroup can refer to either The finite Thompson sporadic group Th studied by John G. Thompson The finite Thompson subgroup...
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include the infinite Thompsongroups T {\displaystyle T} and V {\displaystyle V} . Finitely presented torsion-free infinite simple groups were constructed...
1, 2007, but unlike most candidate exploratory groups, Thompson's organized as a 527 group. Thompson continued to be mentioned as a potential candidate...
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subgroup of index 2 if n is odd, one of which is one of the Thompsongroups. Higman's group is generated by 4 elements a, b, c, d with the relations a...
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