We start by defining the lower central series of a group G as a series of groups G = A_{0}, A_{1}, A_{2}, ..., A_{i}, ..., where each A_{i+1} = [A_{i}, G], the subgroup of G generated by all commutators [x,y] with x in A_{i} and y in G. Thus, A_{1} = [G,G] = G^{1}, the commutator subgroup of G; A_{2} = [G^{1}, 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 natural number n such that A_{n} is trivial. If n is the smallest natural number such that A_{n} 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 nilm group.
For a justification of the term nilpotent, start with a nilpotent group G, an element g of G and define a function f : G → G by f(x) = [x,g]. Then this function is nilpotent in the sense that there exists a natural number n such that f^{n}, the nth iteration of f, sends every element x of G to the identity element.
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 = Z_{0}, Z_{1}, Z_{2}, ..., Z_{i}, ..., where each successive group is defined by:
In this case, Z_{1} is the center of G, and for each successive group, the factor group Z_{i+1}/Z_{i} is the center of G/Z_{i}. For an abelian group, Z_{1} is simply G; a group is called nilpotent of class n if Z_{n} = G for a minimal n.
These two definitions are equivalent: the lower central series reaches the trivial subgroup E if and only if the upper central series reaches G; furthermore, the minimial index n for which this happens is the same in both cases.
As noted above, every abelian group is nilpotent.
For a small nonabelian example, consider the quaternion group Q_{8}. It has center {1, 1} of order 2, and its lower central series is {1}, {1, 1}, Q_{8}; so it is nilpotent of class 2. In fact, every direct sum of finite pgroups is nilpotent.
Since each successive factor group Z_{i+1}/Z_{i} 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:
The last statement 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 sum of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup).
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