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The free abelian group has a polynomial growth rate of order d. The discrete Heisenberg group H 3 {\displaystyle H_{3}} has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Hyman Bass and Yves Guivarch that is discussed in the article on Gromov's theorem .
The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length at most n (relative to a symmetric generating set) is bounded above by a polynomial function p(n). The order of growth is then the least degree of any such polynomial ...
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed by ...
A linear group is not amenable if and only if it contains a non-abelian free group (thus the von Neumann conjecture, while not true in general, holds for linear groups). The Tits alternative is an important ingredient [2] in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result ...
The Breuillard–Green–Tao theorem on classification of approximate groups can be used to give a new proof of Gromov's theorem on groups of polynomial growth.The result obtained is actually a bit stronger since it establishes that there exists a "growth gap" between virtually nilpotent groups (of polynomial growth) and other groups; that is, there exists a (superpolynomial) function such ...
Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The ...
Any group that is virtually abelian. Any nilpotent group. Any semidirect product where N is nilpotent and H is finite. Any semidirect product where N is finite and H is nilpotent. Gromov's theorem says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.