Nilpotent group in the context of "Virtually"

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⭐ Core Definition: Nilpotent group

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}.

Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.

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👉 Nilpotent group in the context of Virtually

In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup such that H has property P.

Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.

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Nilpotent group in the context of Gromov's theorem on groups of polynomial growth

In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.

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LINKED FROM (PARENTS)
← Virtually ← Gromov's theorem on groups of polynomial growth
LINKS TO (CHILDREN)
Mathematics → Group theory → Group (mathematics) → Upper central series → Central series → Lower central series → Abelian group → Solvable group → Finite group → Relatively prime → Order (group theory) → Commutative property → Supersolvable group → Sergei Chernikov →