In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the commutator operation, [x,y]. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.
Definition
We start by defining the lower central series of a group G as a series of groups G = A0, A1, A2, ..., Ai, ..., where each Ai+1 = [Ai, G]. Thus, A1 = [G,G] = G1, the commutator subgroup of G; A2 = [G1, G], etc.
If G is abelian, then [G,G] = E, the trivial subgroup. As an extension of this idea, we call a group G nilpotent if there is some integer n such that An is trivial. If n is the smallest integer such that An is trivial, then we say that G is nilpotent of class n. Every abelian group is nilpotent of class 1. If a group is nilpotent of class at most m, then it is sometimes called a nil-m group.
For a justification of the term nilpotent, note that if we define a function f by way of f(A) = [A,G] for a nil-n group G, then f is then nilpotent as a function, in the sense that f n = 0G, the zero homomorphism of G.
An equivalent definition of a nilpotent group is arrived at by way of the upper central series of G, which is a sequence of groups E = Z0, Z1, Z2, ..., Zi, ..., where each successive group is defined by:
- Zi+1 = {x in G | [x,y] in Zi for all y in G}
In this case, Z1 is the center of G, and for each successive group, the factor group Zi+1/Zi is the center of G/Zi. For an abelian group, Z1 is simply G; a group is called nilpotent of class n if Zn = G for a minimal n.
These two definitions are equivalent; if G is nilpotent of class n, then both the upper central series and lower central series repeat starting at the nth term.
Examples
As noted above, every abelian group is nilpotent.
For a small non-abelian example, consider the quaternion group Q8. It has center {1, -1} of order 2, and its lower central series is {1}, {1, -1}, Q8; so it is nilpotent of class 2. In fact, every direct sum of finite p-groups is nilpotent.
Properties
Since each successive factor group Zi+1/Zi 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:
- G is a nilpotent group.
- If H is a proper normal subgroup of G, then H is a proper normal subgroup of N(H) (the normalizer of H in G)
- Every maximal proper subgroup of G is normal.
- G is the direct sum of its Sylow subgroups.
The last 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 sum of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup).