| Lower Central Series |
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| CATEGORIES ABOUT NILPOTENT GROUP | |
| nilpotent groups | |
| properties of groups | |
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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 = ''G'' , the Subgroup of ''G'' Generated by all commutators with ''x'' in ''A''''i'' and ''y'' in ''G''. Thus, ''A''1 = [''G'',''G'' = ''G''1, the Commutator Subgroup of ''G''; ''A''2 = ''G'' , etc. If ''G'' is abelian, then {Link without Title} = ''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'') = {Link without Title} . 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: |
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