Nilpotent group

The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group :{{unordered list : \{1\} = G_0 \triangleleft G_1 \triangleleft \dots \triangleleft G_n = G where G_{i+1}/G_i \leq Z(G/G_i), or equivalently [G,G_{i+1}] \leq G_i. : G = G_0 \triangleright G_1 \triangleright \dots \triangleright G_n = \{1\} where G_{i+1} = [G_i, G]. : \{1\} = Z_0 \triangleleft Z_1 \triangleleft \dots \triangleleft Z_n = G where Z_1 = Z(G) and Z_{i+1} is the subgroup such that Z_{i+1}/Z_i = Z(G/Z_i). }} 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 n + 1 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. ==Examples==

of the discrete Heisenberg group, a well-known nilpotent group. • As noted above, every abelian group is nilpotent. • For a small non-abelian example, consider the quaternion group Q8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, 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 (proof). For n > 1, the maximal nilpotency class of a group of order pn is n - 1 (for example, a group of order p2 is abelian). 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. infinite nilpotent group. It has nilpotency class 2 with central series 1, Z(H), H. • The multiplicative group of invertible upper triangular n × n matrices over a field F is not in general nilpotent, but is solvable. • Any nonabelian group G such that G/Z(G) is abelian has nilpotency class 2, with central series {1}, Z(G), G. The natural numbers k for which any group of order k is nilpotent have been characterized . ==Explanation of term==

Nilpotent groups are called so because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group G of nilpotence degree n and an element g, the function \operatorname{ad}_g \colon G \to G defined by \operatorname{ad}_g(x) := [g,x] (where [g,x]=g^{-1} x^{-1} g x is the commutator of g and x) is nilpotent in the sense that the nth iteration of the function is trivial: \left(\operatorname{ad}_g\right)^n(x)=e for all x in G. This is not a defining characteristic of nilpotent groups: groups for which \operatorname{ad}_g is nilpotent of degree n (in the sense above) are called n-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 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 is nilpotent of class at most ; in addition, if is a homomorphism of a nilpotent group of class , then the image of is nilpotent of class at most . Statement (d) can be extended to infinite groups: if is a nilpotent group, then every Sylow subgroup of is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in (see torsion subgroup). Many properties of nilpotent groups are shared by hypercentral groups. ==Notes==