Platonov's theorem

The set of Platonow is a theorem from the mathematical field of group theory . From it follow Malcev's theorem and Selberg's lemma . He was proven by Vladimir Petrovich Platonov .

Definitions

Let it be a prime number . A group is a group in which the order of each element is a power of . ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle p}$

A residual -finite group is a group in which for each element there is a surjective group homomorphism on a finite -group with , where denotes the neutral element . ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle g\in G\setminus \left\{1\right\}}$ ${\displaystyle f\colon G\to P}$ ${\displaystyle p}$ ${\displaystyle P}$ ${\displaystyle f(g)\not =e}$ ${\displaystyle e\in P}$

A virtually residual- finite group is a group that contains a subset of finite index that is residual- finite. ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle p}$

Platonov's theorem

Let it be a field and a finitely generated subgroup of the general linear group . ${\displaystyle K}$ ${\displaystyle G\subset GL(n,K)}$

If the characteristic is a prime number, then it is a virtually residual- finite group. ${\displaystyle \mathrm {char} (K)=p}$ ${\displaystyle G}$ ${\displaystyle p}$

If is, then is a virtually residual -finite group for almost all prime numbers . ${\displaystyle \mathrm {char} (K)=0}$ ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle p}$

Applications

Two fundamental and frequently used properties of finitely generated matrix groups follow from Platonow's theorem , namely Malcev's theorem (finitely generated subgroups of are residual finite ) and Selberg's lemma (finitely generated subgroups of are virtually torsion-free ). ${\displaystyle GL(n,\mathbb {C} )}$ ${\displaystyle GL(n,\mathbb {C} )}$

literature

• VP Platonov: A certain problem for finitely generated groups. (Russian) Dokl. Akad. Nauk BSSR 12 (1968) 492-494.
• BAF Wehrfritz: Infinite linear groups. An account of the group-theoretical properties of infinite groups of matrices. Results of mathematics and their border areas, volume 76.Springer-Verlag, New York-Heidelberg, 1973.

Web links

• B. Nica: Linear groups - Malcev's theorem and Selberg's lemma