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] = x^-1y^-1xy. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups. Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only... In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

Definition

We start by defining the lower central series of a group G as a series of groups G = A₀, A₁, A₂, ..., 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, A₁ = [G,G] = G¹, the commutator subgroup of G; A₂ = [G¹, G], etc. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. Note that the set of all commutators of the group is, generally, not a group (in...

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, except for the trivial group, which is nilpotent of class 0. If a group is nilpotent of class at most m, then it is sometimes called a nil-m group. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

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 fⁿ, the n-th 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 = Z₀, Z₁, Z₂, ..., Z_i, ..., where each successive group is defined by:

Z_i+1 = {x in G : [x,y] in Z_i for all y in G}

In this case, Z₁ 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, Z₁ is simply G; a group is called nilpotent of class n if Z_n = G for a minimal n. In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

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 minimal index n for which this happens is the same in both cases.

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. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ...

The discrete Heisenberg group is another example of non-abelian nilpotent group. In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...

Properties

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. In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ...

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. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

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 N(H) (the normalizer of H in G).
• Every maximal proper subgroup of G is normal.
• G is the direct product of its Sylow subgroups.

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). In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... The Sylow theorems of group theory, named after Ludwig Sylow, form a partial converse to Lagranges theorem, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...