Editing Nilpotent group (section)

==Properties==

Since each successive [[factor group]] ''Z''<sub>''i''+1</sub>/''Z''<sub>''i''</sub> in the [[central series|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 ''n'' is nilpotent of class at most ''n'';<ref name="theo7.1.3">Bechtell (1971), p. 51, Theorem 5.1.3</ref> in addition, if ''f'' is a [[group homomorphism|homomorphism]] of a nilpotent group of class ''n'', then the image of ''f'' is nilpotent<ref name="theo7.1.3" /> of class at most ''n''.

The following statements are equivalent for finite groups,<ref>Isaacs (2008), Thm. 1.26</ref> revealing some useful properties of nilpotency:{{ordered list
 | list-style-type=lower-alpha
 | ''G'' is a nilpotent group.
 | If ''H'' is a proper subgroup of ''G'', then ''H'' is a proper [[normal subgroup]] of ''N''<sub>''G''</sub>(''H'') (the [[normalizer]]  of ''H'' in ''G'').  This is called the '''normalizer property''' and can be phrased simply as "normalizers grow".
 | Every [[Sylow subgroup]] of ''G'' is normal.
 | ''G'' is the [[direct product of groups|direct product]] of its Sylow subgroups.
 | If ''d'' divides the [[Order of a group|order]] of ''G'', then ''G'' has a [[normal subgroup]] of order ''d''.
}}

Proof:
; (a)→(b): By induction on |''G''|. If ''G'' is abelian, then for any ''H'', ''N''<sub>''G''</sub>(''H'') = ''G''. If not, if ''Z''(''G'') is not contained in ''H'', then ''h''<sub>''Z''</sub>''H''<sub>''Z''</sub><sup>−1</sup>''h<sup>−1</sup>'' = ''h''''H''''h<sup>−1</sup>'' = ''H'', so ''H''·''Z''(''G'') normalizers ''H''. If ''Z''(''G'') is contained in ''H'', then ''H''/''Z''(''G'') is contained in ''G''/''Z''(''G''). Note, ''G''/''Z''(''G'') is a nilpotent group. Thus, there exists a subgroup of ''G''/''Z''(''G'') which normalizes ''H''/''Z''(''G'')  and ''H''/''Z''(''G'') is a proper subgroup of it. Therefore, pullback this subgroup to the subgroup in ''G'' and it normalizes ''H''. (This proof is the same argument as for ''p''-groups{{snd}}the only fact we needed was if ''G'' is nilpotent then so is ''G''/''Z''(''G''){{snd}}so the details are omitted.)
; (b)→(c): Let ''p''<sub>1</sub>,''p''<sub>2</sub>,...,''p''<sub>''s''</sub> be the distinct primes dividing its order and let ''P''<sub>''i''</sub> in ''Syl''<sub>''p''<sub>''i''</sub></sub>(''G''), 1 ≤ ''i'' ≤ ''s''. Let ''P'' = ''P''<sub>''i''</sub> for some ''i'' and let ''N'' = ''N''<sub>''G''</sub>(''P''). Since ''P'' is a normal Sylow subgroup of ''N'', ''P'' is [[characteristic subgroup|characteristic]] in ''N''. Since ''P'' char ''N'' and ''N'' is a normal subgroup of ''N''<sub>''G''</sub>(''N''), we get that ''P'' is a normal subgroup of ''N''<sub>''G''</sub>(''N''). This means ''N''<sub>''G''</sub>(''N'') is a subgroup of ''N'' and hence ''N''<sub>''G''</sub>(''N'') = ''N''. By (b) we must therefore have ''N'' = ''G'', which gives (c).
; (c)→(d): Let ''p''<sub>1</sub>,''p''<sub>2</sub>,...,''p''<sub>''s''</sub> be the distinct primes dividing its order and let ''P''<sub>''i''</sub> in ''Syl''<sub>''p''<sub>''i''</sub></sub>(''G''), 1 ≤ ''i'' ≤ ''s''. For any ''t'', 1 ≤ ''t'' ≤ ''s'' we show inductively that ''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''</sub> is isomorphic to ''P''<sub>1</sub>×''P''<sub>2</sub>×···×''P''<sub>''t''</sub>. {{paragraph}}Note first that each ''P''<sub>''i''</sub> is normal in ''G'' so ''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''</sub> is a subgroup of ''G''. Let ''H'' be the product ''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''−1</sub> and let ''K'' = ''P''<sub>''t''</sub>, so by induction ''H'' is isomorphic to ''P''<sub>1</sub>×''P''<sub>2</sub>×···×''P''<sub>''t''−1</sub>. In particular,|''H''| = |''P''<sub>1</sub>|&sdot;|''P''<sub>2</sub>|&sdot;···&sdot;|''P''<sub>''t''−1</sub>|. Since |''K''| = |''P''<sub>''t''</sub>|, the orders of ''H'' and ''K'' are relatively prime. Lagrange's Theorem implies the intersection of ''H'' and ''K'' is equal to 1. By definition,''P''<sub>1</sub>''P''<sub>2</sub>···''P''<sub>''t''</sub> = ''HK'', hence ''HK'' is isomorphic to ''H''×''K'' which is equal to ''P''<sub>1</sub>×''P''<sub>2</sub>×···×''P''<sub>''t''</sub>. This completes the induction. Now take ''t'' = ''s'' to obtain (d).
; (d)→(e): Note that a [[p-group]] of order ''p''<sup>''k''</sup> has a normal subgroup of order ''p''<sup>''m''</sup> for all 1≤''m''≤''k''. Since ''G'' is a direct product of its Sylow subgroups, and normality is preserved upon direct product of groups, ''G'' has a normal subgroup of order ''d'' for every divisor ''d'' of |''G''|.
; (e)→(a): For any prime ''p'' dividing |''G''|, the [[Sylow group|Sylow ''p''-subgroup]] is normal. Thus we can apply (c) (since we already proved (c)→(e)).

Statement (d) can be extended to infinite groups: if ''G'' is a nilpotent group, then every Sylow subgroup ''G''<sub>''p''</sub> of ''G'' is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in ''G'' (see [[torsion subgroup]]).

Many properties of nilpotent groups are shared by [[hypercentral group]]s.