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| In [[mathematics]], especially in the fields of [[group theory]] and [[Lie theory]], a '''central series''' is a kind of [[normal series]] of [[subgroup]]s or [[Lie subalgebra]]s, expressing the idea that the commutator is nearly trivial. For [[group (mathematics)|groups]], this is an explicit expression that the group is a [[nilpotent group]], and for [[matrix ring]]s, this is an explicit expression that in some basis the matrix ring consists entirely of [[upper triangular]] matrices with constant diagonal.
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| This article uses the language of group theory; analogous terms are used for Lie algebras. | |
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| The '''lower central series''' and '''upper central series''' (also called the '''descending central series''' and '''ascending central series''', respectively), are [[characteristic series]], which, despite the names, are central series if and only if a group is [[Nilpotent group|nilpotent]].
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| ==Definition==
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| A '''central series''' is a sequence of subgroups
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| :<math>\{1\} = A_0 \triangleleft A_1 \triangleleft \dots \triangleleft A_n = G</math>
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| such that the successive quotients are [[Center (group)|central]]; that is, [''G'', ''A''<sub>''i'' + 1</sub>] ≤ ''A<sub>i</sub>'', where [''G'', ''H''] denotes the [[commutator subgroup]] generated by all ''g''<sup>−1</sup>''h''<sup>−1</sup>''gh'' for ''g'' in ''G'' and ''h'' in ''H''. As [''G'', ''A''<sub>''i'' + 1</sub>] ≤ ''A<sub>i</sub>'' ≤ ''A''<sub>''i'' + 1</sub>, in particular ''A''<sub>''i'' + 1</sub> is normal in ''G'' for each ''i'', and so equivalently we can rephrase the 'central' condition above as: ''A''<sub>''i'' + 1</sub>/''A<sub>i</sub>'' commutes with all of ''G''/''A<sub>i</sub>''.
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| A central series is analogous in [[Lie theory]] to a [[Flag (linear algebra)|flag]] that is strictly preserved by the [[adjoint representation of a Lie group|adjoint action]] (more prosaically, a basis in which each element is represented by a strictly [[upper triangular]] matrix); compare [[Engel's theorem]].
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| A group need not have a central series. In fact, a group has a central series if and only if it is a [[nilpotent group]]. If a group has a central series, then there are two central series whose terms are extremal in certain senses. Since ''A''<sub>1</sub> ≤ ''Z''(''G''), the largest choice for ''A''<sub>1</sub> is precisely ''A''<sub>1</sub> = ''Z''(''G''). Continuing in this way to choose the largest possible ''A''<sub>''i'' + 1</sub> given ''A<sub>i</sub>'' produces what is called the '''upper central series'''. Dually, since ''A<sub>n</sub>''= ''G'', the commutator subgroup [''G'', ''G''] satisfies [''G'', ''G''] = [''G'', ''A<sub>n</sub>''] ≤ ''A''<sub>''n'' − 1</sub>. Therefore the minimal choice for ''A''<sub>''n'' − 1</sub> is [''G'', ''G'']. Continuing to choose ''A<sub>i</sub>'' minimally given ''A''<sub>''i'' + 1</sub> such that [''G'', ''A''<sub>''i'' + 1</sub>] ≤ ''A<sub>i</sub>'' produces what is called the '''lower central series'''. These series can be constructed for any group, and if a group has a central series (is a nilpotent group), these procedures will yield central series.
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| ==Lower central series==
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| The '''lower central series''' (or '''descending central series''') of a group ''G'' is the descending series of subgroups
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| :''G'' = ''G''<sub>1</sub> ⊵ ''G''<sub>2</sub> ⊵ ⋯ ⊵ ''G<sub>n</sub>'' ⊵ ⋯,
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| where each ''G''<sub>''n'' + 1</sub> = [''G<sub>n</sub>'', ''G''], the [[subgroup]] of ''G'' [[generating set of a group|generated]] by all commutators [''x'', ''y''] with ''x'' in ''G<sub>n</sub>'' and ''y'' in ''G''. Thus, ''G''<sub>2</sub> = [''G'', ''G''] = ''G''<sup>(1)</sup>, the [[derived subgroup]] of ''G''; ''G''<sub>3</sub> = <nowiki>[</nowiki><nowiki>[</nowiki>''G'', ''G''], ''G''], etc. The lower central series is often denoted γ<sub>''n''</sub>(''G'') = ''G<sub>n</sub>''.
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| This should not be confused with the '''[[derived series]]''', whose terms are ''G''<sup>(''n'')</sup> := [''G''<sup>(''n''−1)</sup>,''G''<sup>(''n''−1)</sup>], not ''G<sub>n</sub>'' := [''G''<sub>''n''−1</sub>, ''G'']. The series are related by ''G''<sup>(''n'')</sup> ≤ ''G<sub>n</sub>''. In particular, a nilpotent group is a [[solvable group]], and its derived length is logarithmic in its nilpotency class {{harv|Schenkman|1975|p=201,216}}.
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| For infinite groups, one can continue the lower central series to infinite [[ordinal numbers]] via [[transfinite recursion]]: for a [[limit ordinal]] λ, define ''G''<sub>λ</sub> = ∩ { ''G''<sub>α</sub> : α < λ}. If ''G''<sub>λ</sub> = 1 for some ordinal λ, then ''G'' is said to be a '''hypocentral group'''. For every ordinal λ, there is a group ''G'' such that ''G''<sub>λ</sub> = 1, but ''G''<sub>α</sub> ≠ 1 for all α < λ, {{harv|Malcev|1949}}.
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| If ''ω'' is the first infinite ordinal, then ''G''<sub>ω</sub> is the smallest normal subgroup of ''G'' such that the quotient is '''[[residually nilpotent]]''', that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group {{harv|Schenkman|1975|p=175,183}}. In the field of [[combinatorial group theory]], it is an important and early result that [[free group]]s are residually nilpotent. In fact the quotients of the lower central series are free abelian groups with a natural basis defined by '''basic commutators''', {{harv|Hall|1959|loc=Ch. 11}}.
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| If ''G''<sub>ω</sub> = ''G<sub>n</sub>'' for some finite ''n'', then ''G''<sub>ω</sub> is the smallest normal subgroup of ''G'' with nilpotent quotient, and ''G''<sub>ω</sub> is called the '''nilpotent residual''' of ''G''. This is always the case for a finite group, and defines the ''F''<sub>1</sub>(''G'') term in the [[lower Fitting series]] for ''G''.
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| If ''G''<sub>ω</sub> ≠ ''G<sub>n</sub>'' for all finite ''n'', then ''G''/''G''<sub>ω</sub> is not nilpotent, but it is [[residually nilpotent]]. | |
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| There is no general term for the intersection of all terms of the transfinite lower central series, analogous to the hypercenter (below).
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| ==Upper central series==
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| The '''upper central series''' (or '''ascending central series''') of a group ''G'' is the sequence of subgroups
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| :<math>1 = Z_0 \triangleleft Z_1 \triangleleft \cdots \triangleleft Z_i \triangleleft \cdots,</math>
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| where each successive group is defined by:
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| :<math>Z_{i+1} = \{x\in G \mid \forall y\in G:[x,y] \in Z_i \}</math>
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| and is called the '''[[Center (group theory)#Higher centers|''i''th center]]''' of ''G'' (respectively, '''second center''', '''third center''', etc.). In this case, ''Z''<sub>1</sub> is the [[center (group theory)|center]] of ''G'', and for each successive group, the [[factor group]] ''Z''<sub>''i'' + 1</sub>/''Z<sub>i</sub>'' is the center of ''G''/''Z<sub>i</sub>'', and is called an '''upper central series quotient'''.
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| For infinite groups, one can continue the upper central series to infinite [[ordinal numbers]] via [[transfinite recursion]]: for a [[limit ordinal]] λ, define
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| :<math>Z_\lambda(G) = \bigcup_{\alpha < \lambda} Z_\alpha(G).</math>
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| The limit of this process (the union of the higher centers) is called the '''hypercenter''' of the group.
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| If the transfinite upper central series stabilizes at the whole group, then the group is called '''hypercentral'''. Hypercentral groups enjoy many properties of nilpotent groups, such as the '''normalizer condition''' (the normalizer of a proper subgroup properly contains the subgroup), elements of coprime order commute, and [[periodic group|periodic]] hypercentral groups are the [[direct sum of groups|direct sum]] of their [[Sylow subgroup|Sylow ''p''-subgroups]] {{harv|Schenkman|1975|loc=Ch. VI.3}}. For every ordinal λ there is a group ''G'' with ''Z''<sub>λ</sub>(''G'') = ''G'', but ''Z''<sub>α</sub>(''G'') ≠ ''G'' for α < λ, {{harv|Gluškov|1952}} and {{harv|McLain|1956}}.
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| ==Connection between lower and upper central series==
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| There are various connections between the lower central series and upper central series {{harv|Ellis|2001}}, particularly for [[nilpotent group]]s.
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| Most simply, a group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is trivial) if and only if the UCS stabilizes at the first step (the center is the entire group). More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree (and is called the '''nilpotency class''' of the group).
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| However, the LCS stabilizes at the zeroth step if and only if it is [[perfect group|perfect]], while the UCS stabilizes at the zeroth step if and only if it is [[centerless group|centerless]], which are distinct concepts, and show that the lengths of the LCS and UCS need not agree in general.
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| For a perfect group, the UCS always stabilizes by the first step, a fact called [[Grün's lemma]]. However, a centerless group may have a very long lower central series: a noncyclic [[free group]] is centerless, but its lower central series does not stabilize until the first infinite ordinal.
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| == Refined central series ==
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| In the study of [[p-group|''p''-group]]s, it is often important to use longer central series. An important class of such central series are the exponent-''p'' central series; that is, a central series whose quotients are [[elementary abelian group]]s, or what is the same, have [[exponent (group theory)|exponent]] ''p''. There is a unique most quickly descending such series, the lower exponent-''p'' central series λ defined by:
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| :λ<sub>1</sub>(''G'') = ''G'', and
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| :λ<sub>''n'' + 1</sub>(''G'') = [''G'', λ<sub>''n''</sub>(''G'')] (λ<sub>''n''</sub>(''G''))<sup>''p''</sup>
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| The second term, λ<sub>2</sub>(''G''), is equal to [''G'', ''G'']''G<sup>p</sup>'' = Φ(''G''), the [[Frattini subgroup]]. The lower exponent-''p'' central series is sometimes simply called the ''p''-central series.
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| There is a unique most quickly ascending such series, the upper exponent-''p'' central series S defined by:
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| :S<sub>0</sub>(''G'') = 1
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| :S<sub>''n''+1</sub>(''G'')/S<sub>''n''</sub>(''G'') = Ω(Z(''G''/S<sub>''n''</sub>(''G'')))
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| where Ω(''Z''(''H'')) denotes the subgroup generated by (and equal to) the set of central elements of ''H'' of order dividing ''p''. The first term, S<sub>1</sub>(''G''), is the subgroup generated by the minimal normal subgroups and so is equal to the [[socle (mathematics)|socle]] of ''G''. For this reason the upper exponent-''p'' central series is sometimes known as the socle series or even the Loewy series, though the latter is usually used to indicate a descending series.
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| Sometimes other refinements of the central series are useful, such as the Jennings series ''κ'' defined by:
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| :κ<sub>1</sub>(''G'') = ''G'', and | |
| :κ<sub>''n'' + 1</sub>(''G'') = [''G'', κ<sub>''n''</sub>(''G'')] (κ<sub>''i''</sub>(''G''))<sup>''p''</sup>, where ''i'' is the smallest integer larger than or equal to ''n''/''p''.
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| The Jennings series is named after [[S. A. Jennings]] who used the series to describe the Loewy series of the modular [[group ring]] of a ''p''-group.
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| ==See also==
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| * [[Nilpotent series]], an analogous concept for solvable groups
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| ==References==
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| {{refimprove|date=January 2007}}
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| <references/>
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| *{{citation
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| |title=On the Relation between Upper Central Quotients and Lower Central Series of a Group
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| |last=Ellis
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| |first=Graham
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| |journal=Transactions of the American Mathematical Society
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| |volume=353
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| |issue=10
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| |date=October 2001
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| |pages=4219–4234
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| |doi=10.1090/S0002-9947-01-02812-4
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| |jstor=2693793
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| }}
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| *{{Citation | last1=Gluškov | first1=V. M. | title=On the central series of infinite groups | mr=0052427 | year=1952 | journal=Mat. Sbornik N.S. | volume=31 | pages=491–496}}
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| *{{Citation | last1=Hall | first1=Marshall | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan | mr=0103215 | year=1959}}
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| *{{Citation | last1=Malcev | first1=A. I. | author1-link=Anatoly Maltsev | title=Generalized nilpotent algebras and their associated groups | mr=0032644 | year=1949 | journal=Mat. Sbornik N.S. | volume=25 | issue=67 | pages=347–366}}
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| *{{Citation | doi=10.1017/S2040618500033414 | last1=McLain | first1=D. H. | title=Remarks on the upper central series of a group | mr=0084498 | year=1956 | journal=Proc. Glasgow Math. Assoc. | volume=3 | pages=38–44}}
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| *{{Citation | last1=Schenkman | first1=Eugene | title=Group theory | publisher=Robert E. Krieger Publishing | isbn=978-0-88275-070-5 | mr=0460422 | year=1975}}, especially chapter VI.
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| {{DEFAULTSORT:Central Series}}
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| [[Category:Subgroup series]]
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