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Artikel: Nilpotent group GudangMovies21 Rebahinxxi
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In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}.
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 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.
Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.
Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.
Definition
The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group G:
For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G; and G is said to be nilpotent of class n. (By definition, the length is n if there are
n
+
1
{\displaystyle n+1}
different subgroups in the series, including the trivial subgroup and the whole group.)
Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series.
If a group has nilpotency class at most n, then it is sometimes called a nil-n 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 0, and groups of nilpotency class 1 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 Q8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q8; so it is nilpotent of class 2.
The direct product of two nilpotent groups is nilpotent.
All finite p-groups are in fact nilpotent (proof). For n > 1, the maximal nilpotency class of a group of order pn is n - 1 (for example, a group of order p2 is abelian). The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
Furthermore, every finite nilpotent group is the direct product of p-groups.
The multiplicative group of upper unitriangular n × n matrices over any field F is a 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 invertible upper triangular n × n matrices over a field F is not in general nilpotent, but is solvable.
Any nonabelian group G such that G/Z(G) is abelian has nilpotency class 2, with central series {1}, Z(G), G.
The natural numbers k for which any group of order k is nilpotent have been characterized (sequence A056867 in the OEIS).
Explanation of term
Nilpotent groups are called so because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group
G
{\displaystyle G}
of nilpotence degree
n
{\displaystyle n}
and an element
g
{\displaystyle g}
, the function
ad
g
:
G
→
G
{\displaystyle \operatorname {ad} _{g}\colon G\to G}
defined by
ad
g
(
x
)
:=
[
g
,
x
]
{\displaystyle \operatorname {ad} _{g}(x):=[g,x]}
(where
[
g
,
x
]
=
g
−
1
x
−
1
g
x
{\displaystyle [g,x]=g^{-1}x^{-1}gx}
is the commutator of
g
{\displaystyle g}
and
x
{\displaystyle x}
) is nilpotent in the sense that the
n
{\displaystyle n}
th iteration of the function is trivial:
(
ad
g
)
n
(
x
)
=
e
{\displaystyle \left(\operatorname {ad} _{g}\right)^{n}(x)=e}
for all
x
{\displaystyle x}
in
G
{\displaystyle G}
.
This is not a defining characteristic of nilpotent groups: groups for which
ad
g
{\displaystyle \operatorname {ad} _{g}}
is nilpotent of degree
n
{\displaystyle n}
(in the sense above) are called
n
{\displaystyle n}
-Engel groups, and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are 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 Zi+1/Zi in the 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; 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:
Proof:
(a)→(b)
By induction on |G|. If G is abelian, then for any H, NG(H) = G. If not, if Z(G) is not contained in H, then hZHZ−1h−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 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 – the only fact we needed was if G is nilpotent then so is G/Z(G) – so the details are omitted.)
(b)→(c)
Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi(G), 1 ≤ i ≤ s. Let P = Pi for some i and let N = NG(P). Since P is a normal Sylow subgroup of N, P is characteristic in N. Since P char N and N is a normal subgroup of NG(N), we get that P is a normal subgroup of NG(N). This means NG(N) is a subgroup of N and hence NG(N) = N. By (b) we must therefore have N = G, which gives (c).
(c)→(d)
Let p1,p2,...,ps be the distinct primes dividing its order and let Pi in Sylpi(G), 1 ≤ i ≤ s. For any t, 1 ≤ t ≤ s we show inductively that P1P2···Pt is isomorphic to P1×P2×···×Pt. Note first that each Pi is normal in G so P1P2···Pt is a subgroup of G. Let H be the product P1P2···Pt−1 and let K = Pt, so by induction H is isomorphic to P1×P2×···×Pt−1. In particular,|H| = |P1|⋅|P2|⋅···⋅|Pt−1|. Since |K| = |Pt|, 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,P1P2···Pt = HK, hence HK is isomorphic to H×K which is equal to P1×P2×···×Pt. This completes the induction. Now take t = s to obtain (d).
(d)→(e)
Note that a p-group of order pk has a normal subgroup of order pm 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 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 Gp 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
Bechtell, Homer (1971). The Theory of Groups. Addison-Wesley.
Von Haeseler, Friedrich (2002). Automatic Sequences. De Gruyter Expositions in Mathematics. Vol. 36. Berlin: Walter de Gruyter. ISBN 3-11-015629-6.
Hungerford, Thomas W. (1974). Algebra. Springer-Verlag. ISBN 0-387-90518-9.
Isaacs, I. Martin (2008). Finite Group Theory. American Mathematical Society. ISBN 978-0-8218-4344-4.
Palmer, Theodore W. (1994). Banach Algebras and the General Theory of *-algebras. Cambridge University Press. ISBN 0-521-36638-0.
Stammbach, Urs (1973). Homology in Group Theory. Lecture Notes in Mathematics. Vol. 359. Springer-Verlag. review
Suprunenko, D. A. (1976). Matrix Groups. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-1341-2.
Tabachnikova, Olga; Smith, Geoff (2000). Topics in Group Theory. Springer Undergraduate Mathematics Series. Springer. ISBN 1-85233-235-2.
Zassenhaus, Hans (1999). The Theory of Groups. New York: Dover Publications. ISBN 0-486-40922-8.
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In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}.
A direct product of nilpotent groups is obviously nilpotent, and so the “if” direction follows from the preceding corollary. For the converse, let G be a finite nilpotent group.
The direct product of two nilpotent groups is nilpotent. However the analogue of Proposition 2(ii) is not true for nilpotent groups. For example, [S3; S3] = A3 but also [S3; A3] = A3. Here, A3 S3 is the (cyclic) alternating group inside the symmetric group on three letters.
4 days ago · A group G is nilpotent if the upper central sequence 1=Z_0<=Z_1<=Z_2<=...<=Z_n<=... of the group terminates with Z_n=G for some n. Nilpotent groups have the property that each proper subgroup is properly contained in its normalizer.
18 Jul 2020 · A nilpotent group is one of the concepts that is most difficult to grasp, particularly for infinite groups. If G G is a finite nilpotent group then it is just a direct product of p p -groups, and that's normally enough to satisfy yourself.
30 Des 2018 · In any nilpotent group the elements of finite order form a subgroup, the quotient group by which is torsion free. The finitely-generated torsion-free nilpotent groups are exhausted by the groups of integral triangular matrices with 1's along the main diagonal, and their subgroups.
We call (1) a central series of G of length n. The minimal length of a central series is called the nilpotency class of G. For example, an abelian group is nilpotent with nilpotency class · 1. Some trivial observations: First, nilpotent groups are solvable since (2) implies that Gi=Gi+1 is abelian.
Every abelian group is nilpotent. Every finite p-group is nilpotent. Proof. If G is a p-group then so is G/Zi(G) for every i. By Theorem 16.4 if G/Zi(G) is non-trivial then its center Z(G/Zi(G)) a non-trivial group. This means that if Zi(G) %= G then Zi(G) Zi+1(G) we must have Zn(G) = G for some G. and Zi(G) %= Zi+1(G). Since G is finite.
A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps. For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G ; and G is said to be nilpotent of class n.
09 Feb 2018 · Nilpotent groups are related to nilpotent Lie algebras in that a Lie group is nilpotent as a group if and only if its corresponding Lie algebra is nilpotent. The analogy extends to solvable groups as well: every nilpotent group is solvable, because the upper central series is a filtration with abelian quotients.
