In mathematics, specifically group theory, a nilpotentgroupG 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. 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 mathematicianSergei 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 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 :
has a lower central series terminating in the trivial subgroup after finitely many steps. That is, a series of normal subgroups
has an upper central series terminating in the whole group after finitely many steps. That is, a series of normal subgroups
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 . 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.
For a small non-abelian example, consider the quaternion groupQ8, which is a smallest non-abelian p-group. It has center of order 2, and its upper central series is,, Q8; so it is nilpotent of class 2.
The direct product of two nilpotent groups is nilpotent.
All finite p-groups are in fact nilpotent. The maximal class of a group of order pn is n. The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
Furthermore, every finite nilpotent group is the direct product of p-groups.
The multiplicative group of upper unitriangularn x n matrices over any field F is a nilpotent group of nilpotency class n - 1. In particular, taking n = 3 yields the Heisenberg groupH, an example of a non-abelian infinite nilpotent group. It has nilpotency class 2 with central series 1, Z, H.
The multiplicative group of invertible upper triangularn x n matrices over a field F is not in general nilpotent, but is solvable.
Any nonabelian groupG such that G/Z is abelian has nilpotency class 2, with central series, Z, G.
Explanation of term
Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group of nilpotence degree and an element, the function defined by is nilpotent in the sense that the th iteration of the function is trivial: for all in. This is not a defining characteristic of nilpotent groups: groups for which is nilpotent of degree are called -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.
Properties
Since each successive factor groupZi+1/Zi 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 off is nilpotent of class at most n. The following statements are equivalent for finite groups, revealing some useful properties of nilpotency:
G is a nilpotent group.
If H is a proper subgroup of G, then H is a proper normal subgroup of NG. This is called the normalizer property and can be phrased simply as "normalizers grow".
If d divides the order of G, then G has a normal subgroup of order d.
Proof: →: By induction on |G|. If G is abelian, then for any H, NG=G. If not, if Z is not contained in H, then hZHZ−1h−1=h'H'h−1=H, so H·Z normalizers H. If Z is contained in H,then H/Z is contained in G/Z. Note, G/Z is a nilpotent group. Thus, there exists an subgroup of G/Z which normalizers H/Z and H/Z is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup in G and it normalizes H. →: Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi,1≤i≤s. Let P=Pi for some i and let N=NG. Since P is a normal subgroup of N, P is characteristic in N. Since P char N and N is a normal subgroup of NG, we get that P is a normal subgroup of NG. This means NG is a subgroup of N and hence NG=N. By we must therefore have N=G, which gives. →: Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi,1≤i≤s. For any t, 1≤t≤s we show inductively that P1P2…Pt is isomorphic to P1×P2×…×Pt. Note first that each Pi is normal in G so P1P2…Pt is a subgroup of G. Let H be the product P1P2…Pt-1 and let K=Pt,so by inductionH is isomorphic to P1×P2×…×Pt-1. In particular,|H|=|P1|·|P2|·…·|Pt-1|. Since |K|=|Pt|, 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,P1P2…Pt=HK, hence HK is isomorphic to H×K which is equal to P1×P2×…×Pt. This completes the induction. Now take t=s to obtain. →: Note that a P-group of order pk has a normal subgroup of order pm 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|. →: For any prime p dividing |G|, the Sylow p-subgroup is normal. Thus we can apply →). Statement can be extended to infinite groups: if G is a nilpotent group, then every Sylow subgroup Gp of G is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in G. Many properties of nilpotent groups are shared by hypercentral groups.
