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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, a nilpotent group ''G'' is a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
that has an upper central series that terminates with ''G''. Equivalently, its
central series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is of finite length or its
lower central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the Commutator#Group theory, commutator is nearly trivial. For group (mat ...
terminates with . Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having
relatively prime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician
Sergei Chernikov Sergei Nikolaevich Chernikov (11 May 1912 – 23 January 1987; russian: Сергей Николаевич Черников) was a Russian mathematician who contributed significantly to the development of infinite group theory and linear inequaliti ...
. Nilpotent groups arise in
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, as well as in the classification of groups. They also appear prominently in the classification of
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s. Analogous terms are used for
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s (using the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightar ...
) including
nilpotent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, lower central series, and upper central series.


Definition

The definition uses the idea of a
central series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
for a group. The following are equivalent definitions for a nilpotent group : * has a
central series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of finite length. That is, a series of normal subgroups :\ = G_0 \triangleleft G_1 \triangleleft \dots \triangleleft G_n = G :where G_/G_i \leq Z(G/G_i), or equivalently ,G_\leq G_i. * has a
lower central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the Commutator#Group theory, commutator is nearly trivial. For group (mat ...
terminating in the trivial subgroup after finitely many steps. That is, a series of normal subgroups :G = G_0 \triangleright G_1 \triangleright \dots \triangleright G_n = \ :where G_ = _i, G/math>. * has an upper central series terminating in the whole group after finitely many steps. That is, a series of normal subgroups :\ = Z_0 \triangleleft Z_1 \triangleleft \dots \triangleleft Z_n = G :where Z_ = Z(G) and Z_ is the subgroup such that Z_/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

* As noted above, every abelian group is nilpotent. * For a small non-abelian example, consider the
quaternion group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''Q''8, which is a smallest non-abelian ''p''-group. It has center of order 2, and its upper central series is , , ''Q''8; so it is nilpotent of class 2. * The
direct productIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of two nilpotent groups is nilpotent. * All finite ''p''-groups are in fact nilpotent (
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
). The maximal class of a group of order ''p''''n'' is ''n'' (for example, any group of order 2 is nilpotent of class 1). The 2-groups of maximal class are the generalised
quaternion group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, the dihedral groups, and the semidihedral groups. * Furthermore, every finite nilpotent group is the direct product of ''p''-groups. * The multiplicative group of upper Triangular matrix#Unitriangular matrix, unitriangular ''n'' x ''n'' matrices over any field ''F'' is a Unipotent algebraic group, nilpotent group of nilpotency class ''n'' - 1. In particular, taking ''n'' = 3 yields the Heisenberg group ''H'', an example of a non-abelian infinite nilpotent group. It has nilpotency class 2 with central series 1, ''Z''(''H''), ''H''. * The multiplicative group of Borel subgroup, invertible upper triangular ''n'' x ''n'' matrices over a field ''F'' is not in general nilpotent, but is solvable group, solvable. * Any nonabelian group ''G'' such that ''G''/''Z''(''G'') is abelian has nilpotency class 2, with central series , ''Z''(''G''), ''G''.


Explanation of term

Nilpotent groups are so called 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_g \colon G \to G defined by \operatorname_g(x) := [g,x] (where [g,x]=g^ x^ 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_g\right)^n(x)=e for all x in G. This is not a defining characteristic of nilpotent groups: groups for which \operatorname_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 (group theory), order, and are conjectured to be nilpotent as long as they are Generating set of a group, 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 ''Z''''i''+1/''Z''''i'' 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'';Bechtell (1971), p. 51, Theorem 5.1.3 in addition, if ''f'' is a group homomorphism, 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,Isaacs (2008), Thm. 1.26 revealing some useful properties of nilpotency: *(a) ''G'' is a nilpotent group. *(b) If ''H'' is a proper subgroup of ''G'', then ''H'' is a proper normal subgroup of ''N''''G''(''H'') (the normalizer of ''H'' in ''G''). This is called the normalizer property and can be phrased simply as "normalizers grow". *(c) Every Sylow subgroup of ''G'' is normal. *(d) ''G'' is the direct product of groups, direct product of its Sylow subgroups. *(e) 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''''G''(''H'')=''G''. If not, if ''Z''(''G'') is not contained in ''H'', then ''h''''Z''''H''''Z''−1''h−1''=''h'H'h−1''=''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 an subgroup of ''G''/''Z''(''G'') which normalizers ''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''-groupsthe only fact we needed was if ''G'' is nilpotent then so is ''G''/''Z''(''G'')so the details are omitted.) (b)→(c): Let ''p''1,''p''2,...,''p''''s'' be the distinct primes dividing its order and let ''P''''i'' in ''Syl''''p''''i''(''G''),1≤''i''≤''s''. Let ''P''=''P''''i'' for some ''i'' and let ''N''=''N''''G''(''P''). Since ''P'' is a normal subgroup of ''N'', ''P'' is characteristic in ''N''. Since ''P'' char ''N'' and ''N'' is a normal subgroup of ''N''''G''(''N''), we get that ''P'' is a normal subgroup of ''N''''G''(''N''). This means ''N''''G''(''N'') is a subgroup of ''N'' and hence ''N''''G''(''N'')=''N''. By (b) we must therefore have ''N''=''G'', which gives (c). (c)→(d): Let ''p''1,''p''2,...,''p''''s'' be the distinct primes dividing its order and let ''P''''i'' in ''Syl''''p''''i''(''G''),1≤''i''≤''s''. For any ''t'', 1≤''t''≤''s'' we show inductively that ''P''1''P''2...''P''''t'' is isomorphic to ''P''1×''P''2×...×''P''''t''. Note first that each ''P''''i'' is normal in ''G'' so ''P''1''P''2...''P''''t'' is a subgroup of ''G''. Let ''H'' be the product ''P''1''P''2...''P''''t-1'' and let ''K''=''P''''t'', so by induction ''H'' is isomorphic to ''P''1×''P''2×...×''P''''t-1''. In particular,, ''H'', =, ''P''1, ·, ''P''2, ·...·, ''P''''t-1'', . Since , ''K'', =, ''P''''t'', , 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''1''P''2...''P''''t''=''HK'', hence ''HK'' is isomorphic to ''H''×''K'' which is equal to ''P''1×''P''2×...×''P''''t''. This completes the induction. Now take ''t''=''s'' to obtain (d). (d)→(e): Note that a P-group of order ''p''''k'' has a normal subgroup of order ''p''''m'' 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''''p'' 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 groups.


Notes


References

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review
* * * {{DEFAULTSORT:Nilpotent Group Nilpotent groups Properties of groups