Nilpotente group

Nilpotente group is a term from the field of group theory , a branch of mathematics . In a certain sense he generalizes the concept of the commutative group “as little as possible” for finite groups : every commutative group is nilpotent, but not vice versa. Finite commutative groups (except for isomorphism) can be clearly represented as a direct product of a finite number of cyclic groups of prime power order. This is a statement of the main theorem about finitely generated Abelian groups . In the case of finite nilpotent groups, the p -Sylow groups take on the role of the cyclic groups: Every finite nilpotent group is (except for isomorphism) a direct product of its p -Sylow groups. The definition of the term “nilpotent group” is based on the more general concept of a chain of subgroups (with certain properties), which is explained in the article “ Series (group theory) ”.

Characterizations

Various equivalent characterizations can be given for nilpotent groups. They are often introduced by looking at certain series . Define the commutators inductively for a group

${\displaystyle L_{1}(G)=G}$

${\displaystyle L_{n}(G):=[L_{n-1}(G),G]}$for . ${\displaystyle n>1}$

This gives the descending central row

${\displaystyle G=L_{1}(G)\geq L_{2}(G)\geq \dots \geq L_{n}(G)}$.

It is called nilpotent if the descending central sequence for one ends in the one group . ${\displaystyle G}$ ${\displaystyle n\in \mathbb {N} }$

Similarly, one can define inductively for the -th center as follows. ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle Z_{n}(G)}$

${\displaystyle Z_{0}(G):=1}$,

${\displaystyle Z_{1}(G):=Z(G)}$

${\displaystyle Z_{n}(G)}$is the archetype of . ${\displaystyle Z(G/Z_{n-1}(G))}$

So is

${\displaystyle 1=Z_{0}(G)\leq Z_{1}(G)\leq \dots \leq Z_{n}(G)}$

an ascending row; the ascending central row . One can show that if and only if this series rises up to completely is nilpotent and that the lengths of both chains are the same, which leads to the definition of the nilpotency class (also nilpotency degree ). The degree of nilpotency is exactly the common length of these two series. ${\displaystyle G}$ ${\displaystyle G}$

The following characterizations apply to finite groups:

• All -Sylow subsets are normal in . In particular, is a direct product of their -Sylow subgroups. ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle p}$
• For prime numbers , products of elements are again elements. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$
• Every subgroup of is subnormal . ${\displaystyle G}$
• For different prime numbers and the commutators of elements with elements are equal to the neutral element. ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$
• If is a real subgroup of , then real is included in its normalizer. ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle U}$
• If there is a maximum subgroup, then normal is in . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle G}$

properties

• Subgroups, factor groups, and homomorphic images of a nilpotent group are nilpotent.
• Conversely , if there is a nilpotent normal divisor and also nilpotent, then in general it is not nilpotent. One example is the non-nilpotent group S ₃ , which has a normal divisor that is isomorphic to the cyclic and thus nilpotent group and whose factor group is also nilpotent. But the following theorem applies: ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle G}$ ${\displaystyle \mathbb {Z} _{3}}$ ${\displaystyle N}$ ${\displaystyle S_{3}/N\cong \mathbb {Z} _{2}}$
• Philip Hall : If there is a group with a nilpotent normal divisor , so that is nilpotent, then is also nilpotent. Where the commutator group is from . ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle G/N'}$ ${\displaystyle G}$ ${\displaystyle N'}$ ${\displaystyle N}$
• Every nilpotent group can be resolved . The converse is generally wrong, as evidenced by the symmetrical group S ₃ .
• Nilpotent groups that are finally generated can be resolved ; the converse does not apply here either.
• Products of nilpotent normal factors in a group are nilpotent. This property leads to the definition of the fitting subgroup , (according to Hans Fitting ) the product of all nilpotent normal factors.

classification

• The direct product of nilpotent groups is nilpotent if the nilpotency degrees of the factors are limited.
• Every finite p -group is nilpotent. An infinite p -group is nilpotent if the order of the group elements is restricted. (Note that this requirement is stronger than the requirement of finite order for group elements, which is already guaranteed by the definition of the p -group.)
• A finite nilpotent group is isomorphic to the direct product of its p -Sylow subgroups. Note that every nilpotent group has exactly one (possibly trivial) p -Sylow subgroup for every prime p .

Examples

• A non-trivial group is nilpotent of degree 1 if and only if it is Abelian .
• Let it be a body and a natural number. The set of n × n matrices of the form ${\displaystyle K}$ ${\displaystyle n}$

${\displaystyle {\begin{pmatrix}1&*&\cdots &*\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &*\\0&\cdots &0&1\end{pmatrix}}}$(the stars stand for any elements of ) ${\displaystyle K}$

is a subgroup of the group of invertible n × n matrices , the group of strict upper triangular matrices . It is nilpotent with a degree of nilpotency . In a special case , this group also bears the name Heisenberg Group . ${\displaystyle n-1}$
${\displaystyle n=3}$ ${\displaystyle K=\mathbb {R} }$

• The dihedral group with elements is nilpotent if and only if ; in this case the degree of nilpotency is the same . ${\displaystyle D_{n}}$ ${\displaystyle n}$ ${\displaystyle n=2^{r}}$ ${\displaystyle r-1}$
• The Frattini group is always nilpotent and, if nilpotent, then also . ${\displaystyle \Phi (G)}$ ${\displaystyle G/\Phi (G)}$ ${\displaystyle G}$

literature

• Thomas W. Hungerford: Algebra (= Graduate Texts in Mathematics. Vol. 73). 5 ^th printing. Springer, New York NY a. a. 1989, ISBN 0-387-90518-9 .

Individual evidence

1. ^ Michael Aschbacher : Finite group theory , Cambridge Studies in Advanced Mathematics, Volume 10 , 2nd edition, Cambridge University Press (2000), ISBN 0-521-78145-0 , pp. 28-29.
2. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.2.4
3. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.2.10
4. ↑ Hans Kurzweil , Bernd Stellmacher : The theory of finite groups. An introduction. Springer, New York a. a. 2004, ISBN 0-387-40510-0 , p. 105.