NILPOTENT GROUP

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''^-1''y''^-1''xy''. Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.

Contents

Definition

Examples

Properties

Definition

We start by defining the 'lower' (or 'descending') '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, 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.
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 ''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' (or 'ascending') '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 centre of ''G'', and for each successive group, the factor group ''Z''_''i''+1/''Z''_''i'' is the centre 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 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 ''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 product of finitely many finite ''p''-groups is nilpotent.
The discrete Heisenberg group is another example of non-abelian nilpotent group.

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.
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 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).