Nilpotent group

A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with {1}. A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with {1}. In group theory, 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. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group G: For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G; and G is said to be nilpotent of class n. (By definition, the length is n if there are n + 1 {displaystyle n+1} different subgroups in the series, including the trivial subgroup and the whole group.) Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series.If a group has nilpotency class at most n, then it is sometimes called a nil-n group. It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly the non-trivial abelian groups. Nilpotent groups are so called because the 'adjoint action' of any element is nilpotent, meaning that for a nilpotent group G {displaystyle G} of nilpotence degree n {displaystyle n} and an element g {displaystyle g} , the function ad g : G → G {displaystyle operatorname {ad} _{g}colon G o G} defined by ad g ⁡ ( x ) := [ g , x ] {displaystyle operatorname {ad} _{g}(x):=} (where [ g , x ] = g − 1 x − 1 g x {displaystyle =g^{-1}x^{-1}gx} is the commutator of g {displaystyle g} and x {displaystyle x} ) is nilpotent in the sense that the n {displaystyle n} th iteration of the function is trivial: ( ad g ) n ( x ) = e {displaystyle left(operatorname {ad} _{g} ight)^{n}(x)=e} for all x {displaystyle x} in G {displaystyle G} .