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In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted , which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to von Neumann conjecture.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 which 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 () 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. introduced an infinite family of finitely presented simple groups, including Thompson's group ''V'' as a special case. ==Presentations== A finite presentation of ''F'' is given by the following expression: : where () is the usual group theory commutator, ''xyx''−1''y''−1. Although ''F'' has a finite presentation with 2 generators and 2 relations, it is most easily and intuitively described by the infinite presentation: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Thompson groups」の詳細全文を読む スポンサード リンク
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