Local group

Local groups are investigated in the mathematical sub-area of group theory. It is a generalization of finite groups in such a way that only the finiteness of each finitely generated subgroup is required, the group itself can be infinite . (The Separated letters locally finite is found in the literature.)

definition

A group is called locally finite if every subgroup generated by finitely many elements is finite.

An apparently equivalent formulation is: A group is called locally finite if every finite subset is contained in a finite subgroup.

Examples

• Finite groups are locally finite.
• The groups of examiners are infinite, but locally finite.
• Every solvable torsion group is locally finite.
• Be an infinite amount. Then the group of all permutations on , which leave all but a finite number of points out fixed, is locally finite. With this one can construct locally finite groups of arbitrary thickness . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$
• According to a theorem of Issai Schur , every torsion subgroup of the general linear group over a finite-dimensional vector space is a locally finite group.

Counterexamples

• Since locally finite groups are torsion groups, because every element is by definition in a finite group, every non-torsion group is a counterexample. So all groups with an element of infinite order are not locally finite, in particular the additive group of integers is not locally finite.
• Tarski groups are torsion groups that are not locally finite.

Inheritance properties

• Subgroups of locally finite groups are again locally finite.
• Quotient groups of locally finite groups are again locally finite.
• Group extensions of locally finite groups are again locally finite, i. i.e., is a normal divisor and are and locally finite, so too . ${\displaystyle N\subset G}$ ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle G}$
• The restricted direct product of finite groups is locally finite. So is a family of finite groups, so is ${\displaystyle (G_{i})_{i\in I}}$

${\displaystyle {\hat {\prod _{i\in I}}}G_{i}:=\{(x_{i})_{i\in I}\mid x_{i}\in G_{i}{\text{ für alle }}i\in I,x_{i}=1_{i}{\text{ für fast alle }}i\in I\}\subset \prod _{i\in I}G_{i}}$

local finite, where the neutral element is in . ${\displaystyle 1_{i}}$ ${\displaystyle G_{i}}$

Sylow groups

As in the theory of finite groups, p -Sylow groups are maximal p -subgroups of a group, where let be a prime number . A standard application of Zorn's lemma shows that every group, even an infinite one, has -Sylow groups. This raises the question of whether two -Sylowgruppen as in the finite case conjugated are. This is generally not the case, even for countable locally finite groups. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

As an example, consider the restricted, countable product of the symmetric group S ₃

${\displaystyle G={\hat {\prod _{i\in \mathbb {N} }}}S_{3}}$.

For each let an element of order 2. Then we can show that each subgroup ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle x_{i}\in S_{3}}$

${\displaystyle U:={\hat {\prod _{i\in I}}}\langle x_{i}\rangle \subset G}$

is a 2-Sylow group, where the two-element subgroup created by is. For each one has three possible choices of the , so that there are uncountably many 2-Sylow groups. They cannot all be conjugated, because a conjugation is mediated by a group element and there are only countably many of them. The conjugation of all 2-Sylow groups therefore fails for reasons of power . But that is the only possible reason, because the following sentence applies: ${\displaystyle \langle x_{i}\rangle }$ ${\displaystyle x_{i}\in S_{3}}$ ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle x_{i}}$

• Let it be a countable, locally finite group and a prime number. All -Sylow groups are conjugated with one another if and only if there are at most a countable number of them. ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle p}$

Abelian subgroups

In group theory there was the old question, the exact origin of which seems unclear, whether an infinite group always contains an infinite Abelian group. It turns out that this is generally not the case. The Tarski groups are extreme counterexamples, because they are not Abelian themselves and every real, non-trivial subgroup is finite in the prime order. P. Hall , CR Kulatilaka and MI Kargapolow received a positive answer for local groups:

• Every infinite, locally finite group contains an infinite Abelian group.

The proof uses Feit-Thompson's theorem . There is no known evidence that this tool can be used.

Individual evidence

1. ↑ Sergei Nikolajewitsch Tschernikow : Finiteness conditions in group theory , Deutscher Verlag der Wissenschaften (1963)
2. ^ Wilhelm Specht : Group Theory , Springer-Verlag 1956, Basic Teachings of Mathematical Sciences , ISBN 978-3-642-94668-4 , Chapter 1.4.5 Local Group Properties , Definition 10
3. ^ B. Hartley, GM Seitz, AV Borovik, RM Bryant: Finite and Locally Finite Groups , Springer-Verlag 1995, ISBN 978-94-010-4145-4 , Introduction
4. ↑ OH cone , BAF Wehrfritz: Locally Finite Groups , North Holland Publishing Company (1973), ISBN 0-7204-2454-2 , Example II on page 9
5. ^ Martyn R. Dixon: Sylow theory, formation and fitting classes in locally finite groups , World Scientific Publishing (1994), ISBN 9-8102-1795-1 , Theorem 1.4.16
6. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 14.3.1
7. ^ Martyn R. Dixon: Sylow theory, formation and fitting classes in locally finite groups , World Scientific Publishing (1994), ISBN 9-8102-1795-1 , example 1.4.4
8. ↑ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , example in chap. 14.3
9. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 14.3.6
10. ↑ P. Hall, CR Kulatilaka: A property of locally finite groups ., J. London Math Soc. (1964) Volume 39, Pages 235-239
11. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 14.3.7
12. ↑ M. Kargapolov: On a problem of O. Ju. Schmidt , Sib. Mat. Zh. (1963), Volume 4, Pages 232-235