Fitting subgroup

The fitting subgroup , named after Hans Fitting , is a construction of a certain subgroup considered in the mathematical branch of group theory , in many cases it is the largest nilpotent subgroup .

definition

It is a group. The subgroup created by all nilpotent normal subgroups is called the fitting subgroup and is denoted by or . ${\displaystyle G}$ ${\displaystyle \mathrm {Fit} (G)}$ ${\displaystyle \mathrm {F} (G)}$

Caution: The name could be confused with the Frattini group , but the latter is also referred to by many authors . ${\displaystyle \mathrm {F} (G)}$ ${\displaystyle \Phi (G)}$

Remarks

The fitting subgroup is always a normal divisor, even a characteristic subgroup . In general it is not nilpotent itself (see examples below), but the following applies

• Fitting's theorem : If and are nilpotent normal divisors of a group, then its complex product is also a nilpotent normal divisor. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle MN}$

If it is finitely generated, then from Fitting's theorem it immediately follows that is nilpotent, because then this subgroup is a finite complex product of nilpotent normal subgroups. This applies in particular to groups with maximum conditions, that is to say to groups in which every non-empty family of subgroups has a maximum element, because in such groups every subgroup is finitely generated. In particular, the fitting subgroup of a finite group is always the largest nilpotent normal subgroup contained therein. ${\displaystyle \mathrm {Fit} (G)}$ ${\displaystyle \mathrm {Fit} (G)}$

The fitting subgroup can be trivial . For finite groups this is exactly the case for the semi-simple groups .

Examples

• For nilpotent groups is by definition ${\displaystyle G}$ ${\displaystyle \mathrm {Fit} (G)=G}$
• If it is simple and non-Abelian , it is , because the group is not nilpotent and there are no real normal dividers. ${\displaystyle G}$ ${\displaystyle \mathrm {Fit} (G)=\{1\}}$
• For the symmetric group S ₃ is . ${\displaystyle \mathrm {Fit} (S_{3})=\{(1),(123),(132)\}}$
• For solvable groups is , because the smallest non-trivial derived group is an Abelian and thus nilpotent normal divisor and as such is contained in the fitting subgroup. ${\displaystyle G}$ ${\displaystyle \mathrm {Fit} (G)\not =\{1\}}$
• Let be a sequence of nilpotent groups of nilpotence class n . Then the direct sum is not nilpotent, but it does hold , in particular we have hereby an example of a nilpotent fitting subgroup. ${\displaystyle H_{n}}$ ${\displaystyle \textstyle G:=\bigoplus _{n\in \mathbb {N} }H_{n}}$ ${\displaystyle \mathrm {Fit} (G)=G}$

Finite groups

• In finite groups, the fitting subgroup is the average of the centralizers of the main factors.

Regarding the terms used here

${\displaystyle \{1\}=G_{0}\vartriangleleft G_{1}\vartriangleleft \ldots \vartriangleleft G_{n}=G}$

a main series, i.e. a normal series that cannot be further refined. The factor groups are called main factors and their centralizers are ${\displaystyle G_{i+1}/G_{i}}$

${\displaystyle C_{G}(G_{i+1}/G_{i}):=\{x\in G|\,\forall y\in G_{i+1}:[x,y]\in G_{i}\}}$.

The above statement means

${\displaystyle \mathrm {Fit} (G)=\bigcap _{i=0}^{n-1}C_{G}(G_{i+1}/G_{i})}$.

For a prime number be the average of all p -Sylow groups . If further denotes the set of all prime numbers, then applies ${\displaystyle p}$ ${\displaystyle K_{p}}$ ${\displaystyle \mathbb {P} }$

• For a finite group is , ${\displaystyle G}$ ${\displaystyle \mathrm {Fit} (G)=\bigoplus _{p\in \mathbb {P} ,p|\mathrm {ord} (G)}K_{p}}$

where means that p shares the group order . ${\displaystyle p|\mathrm {ord} (G)}$

For finite groups there is the following relation to the Frattini group : ${\displaystyle \Phi (G)}$

• ${\displaystyle [\mathrm {Fit} (G),\mathrm {Fit} (G)]\leq \Phi (G)\leq \mathrm {Fit} (G)}$
• ${\displaystyle \mathrm {Fit} (G)/\Phi (G)=\mathrm {Fit} (G/\Phi (G))}$.

The length of the Nile power

Using the fitting subgroup, one can recursively create the so-called upper nilpotent series as follows

${\displaystyle \{1\}=U_{0}(G)\leq U_{1}(G)\leq \ldots }$

form a group . One sets ${\displaystyle G}$

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

${\displaystyle U_{i+1}(G)/U_{i}(G):=\mathrm {Fit} (G/U_{i}(G))}$.

This series eventually reached , it is called the smallest with the Nilpotenzlänge of . This is always the case for solvable groups and the nilpower length is the smallest number for which there is a series ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle U_{n}(G)=G}$ ${\displaystyle G}$ ${\displaystyle n}$

${\displaystyle \{1\}=G_{0}\vartriangleleft G_{1}\vartriangleleft \ldots \vartriangleleft G_{n}=G}$

from normal parts , so that the factors are nilpotent. ${\displaystyle G_{i}\vartriangleleft G}$ ${\displaystyle G_{i+1}/G_{i}}$

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.2.8
2. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , page 133: The Fitting Subgroup
3. ^ WR Scott: Group Theory , Dover Publications (2010), ISBN 978-0-486-65377-8 , Chapter 7.4: Fitting Subgroup + Exercise 9.2.32
4. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.2.9
5. ^ JC Lennox, DJS Robinson: The Theory of Infinite Soulble Groups , Clarendon Press - Oxford 2004, ISBN 0-19-850728-3 , page 9
6. ^ WR Scott: Group Theory , Dover Publications (2010), ISBN 978-0-486-65377-8 , sentence 7.4.3
7. ^ W. Keith Nicholson: Introduction to Abstract Algebra , John Wiley & Sons Inc (2000), ISBN 978-1-118-13535-8 , Chapter 9.3, Theorem 10
8. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 5.4.5