Nilpotent group

Equivalent definitions in tabular format
The smallest possible $$c$$ for all definitions is termed the nilpotency class (sometimes written nilpotence class) of $$G$$. We usually say a group is of nilpotency class $$c$$ if its nilpotency class is less than or equal to $$c$$.

Examples

 * The trivial group is nilpotent, of nilpotency class zero.
 * Any abelian group is nilpotent, of nilpotency class one (note that the nilpotency class is exactly one for nontrivial abelian groups).
 * Any group of prime power order is nilpotent. . Thus, exploring groups of prime power order is a good starting point for exploring nilpotent groups. See groups of order 8, groups of order 16, groups of order 32, groups of order 27, groups of order 81. See also nilpotency class distribution of finite p-groups.
 * The dihedral group of order 8 is the smallest (in terms of order) nilpotent group which is not abelian. It is a group of nilpotency class two.
 * The quaternion group is also the smallest (in terms of order) nilpotent group which is not abelian. This also has order eight.
 * For $$p \ne 2$$, there are in general two non-abelian groups of order $$p^3$$, both of which are nilpotent (of class two): semidirect product of cyclic group of prime-square order and cyclic group of prime order and unitriangular matrix group:UT(3,p).

Conjunction with other properties
Conjunctions with other group properties:

Conjunctions with subgroup properties:


 * Nilpotent normal subgroup
 * Nilpotent characteristic subgroup
 * Nilpotent subnormal subgroup

Testing
To test whether a given group is nilpotent or not using GAP, enter:

IsNilpotentGroup (group);

where group is either the definition of a group or a name for a group already defined.

The class of all nilpotent groups is specified as NilpotentGroups.

Formalisms
A group is nilpotent if and only if the ascending series corresponding to the center subgroup-defining function (which is the upper central series) terminates at the whole group in finitely many steps.

A group $$G$$ is nilpotent if and only if the diagonal subgroup $$\{ (g,g) \mid g \in G \}$$ is subnormal in the group $$G \times G$$. In fact, the nilpotency class of $$G$$ equals the subnormal depth of the diagonal subgroup.

Study of this notion
While 20F18 is the subject class used for nilpotent groups, the subject class used for finite nilpotent groups in particular is 20D15.

Closely related is 20F19: Generalizations of nilpotent and solvable groups.