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, its central series is 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.
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.
Definition.
The definition uses the idea of a central series for a group. 
The following are equivalent definitions for a nilpotent group :
For a nilpotent group, the smallest such that has a central series of length is called the nilpotency class of ; and is said to be nilpotent of class . (By definition, the length is if there are formula_10 different subgroups in the series, including the trivial subgroup and the whole group.)
Equivalently, the nilpotency class of equals the length of the lower central series or upper central series.
If a group has nilpotency class at most , then it is sometimes called a nil- 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 , and groups of nilpotency class  are exactly the non-trivial abelian groups.
Explanation of term.
Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group formula_11 of nilpotence degree formula_12 and an element formula_13, the function formula_14 defined by formula_15 (where formula_16 is the commutator of formula_13 and formula_18) is nilpotent in the sense that the formula_12th iteration of the function is trivial: formula_20 for all formula_18 in formula_11.
This is not a defining characteristic of nilpotent groups: groups for which formula_23 is nilpotent of degree formula_12 (in the sense above) are called formula_12-Engel groups, and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.
An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).
Properties.
Since each successive factor group "Z""i"+1/"Z""i" in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.
Every subgroup of a nilpotent group of class "n" is nilpotent of class at most "n"; in addition, if "f" is a homomorphism of a nilpotent group of class "n", then the image of "f" is nilpotent of class at most "n".
The following statements are equivalent for finite groups, revealing some useful properties of nilpotency:
Proof:
(b)→(c): Let "p"1,"p"2...,"p""s" be the distinct primes dividing its order and let "P""i" in "Syl""p""i"("G"),1≤"i"≤"s". Let "P"="P""i" for some "i" and let "N"="N""G"("P"). Since "P" is a normal subgroup of "N", "P" is characteristic in "N". Since "P" char "N" and "N" is a normal subgroup of "N""G"("N"), we get that "P" is a normal subgroup of "N""G"("N"). This means "N""G"("N") is a subgroup of "N" and hence "N""G"("N")="N". By (b) we must therefore have "N"="G", which gives (c).
(c)→(d): Let "p"1,"p"2...,"p""s" be the distinct primes dividing its order and let "P""i" in "Syl""p""i"("G"),1≤"i"≤"s". For any "t", 1≤"t"≤"s" we show inductively that "P"1"P"2..."P""t" is isomorphic to "P"1×"P"2×...×"P""t".
Note first that each "P""i" is normal in "G" so "P"1"P"2..."P""t" is a subgroup of "G". Let "H" be the product "P"1"P"2..."P""t-1" and let "K"="P""t",so by induction "H" is isomorphic to "P"1×"P"2×...×"P""t-1". In particular,|"H"|=|"P"1|·|"P"2|·...·|"P""t-1"|. Since |"K"|=|"P""t"|, the orders of "H" and "K" are relatively prime. Lagrange's Theorem implies the intersection of "H" and "K" is equal to 1. By definition,"P"1"P"2..."P""t"="HK", hence "HK" is isomorphic to "H"×"K" which is equal to "P"1×"P"2×...×"P""t". This completes the induction. Now take "t"="s" to obtain (d).
(d)→(e): Note that a P-group of order "p""k" has a normal subgroup of order "p""m" for all 1≤"m"≤"k". Since "G" is a direct product of its Sylow subgroups, and normality is preserved upon direct product of groups, "G" has a normal subgroup of order "d" for every divisor "d" of |"G"|.
(e)→(a): For any prime "p" dividing |"G"|, the Sylow "p"-subgroup is normal. Thus we can apply (c) (since we already proved (c)→(e)).
Statement (d) can be extended to infinite groups: if "G" is a nilpotent group, then every Sylow subgroup "G""p" of "G" is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in "G" (see torsion subgroup).
Many properties of nilpotent groups are shared by hypercentral groups.