Namespaces
Variants
Actions

Difference between revisions of "Locally nilpotent group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Tex done)
 
Line 1: Line 1:
A group in which every finitely-generated subgroup is nilpotent (see [[Nilpotent group|Nilpotent group]]; [[Finitely-generated group|Finitely-generated group]]). In a locally nilpotent group the elements of finite order form a [[Normal subgroup|normal subgroup]], the torsion part of this group (cf. [[Periodic group|Periodic group]]). This subgroup is the direct product of its Sylow subgroups, and the quotient group with respect to it is torsion-free. A locally nilpotent torsion-free group (cf. [[Group without torsion|Group without torsion]]) has the uniqueness-of-roots property: If, for elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604802.png" /> and any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604803.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604804.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604805.png" />. Every locally nilpotent torsion-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604806.png" /> has a Mal'tsev completion, that is, it can be imbedded in a unique locally nilpotent torsion-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604807.png" /> such that all equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604808.png" /> are solvable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l0604809.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048011.png" /> is any element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048012.png" />. This completion is functorial, that is, any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048013.png" /> of locally nilpotent torsion-free groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048015.png" /> can be uniquely extended to a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060480/l06048016.png" />.
+
A group in which every finitely-generated subgroup is nilpotent (see [[Nilpotent group]]; [[Finitely-generated group]]). In a locally nilpotent group the elements of finite order form a [[normal subgroup]], the torsion part of this group (cf. [[Periodic group]]). This subgroup is the direct product of its [[Sylow subgroup]]s, and the quotient group with respect to it is torsion-free. A locally nilpotent torsion-free group (cf. [[Group without torsion]]) has the uniqueness-of-roots property: If, for elements $a$ and $b$ and any integer $n\ne0$, one has $a^n = b^n$, then $a=b$. Every locally nilpotent torsion-free group $G$ has a Mal'tsev completion, that is, it can be imbedded in a unique locally nilpotent torsion-free group $G^*$ such that all equations of the form $x^n=g$ are solvable in $G^*$, where $n \ne 0$ and $g$ is any element of $G$. This completion is functorial, that is, any homomorphism $f : G_1 \rightarrow G_2$ of locally nilpotent torsion-free groups $G_1$ and $G_2$ can be uniquely extended to a homomorphism $f^* : G_1^* \rightarrow G_2^*$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,  J.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 18:28, 5 April 2018

A group in which every finitely-generated subgroup is nilpotent (see Nilpotent group; Finitely-generated group). In a locally nilpotent group the elements of finite order form a normal subgroup, the torsion part of this group (cf. Periodic group). This subgroup is the direct product of its Sylow subgroups, and the quotient group with respect to it is torsion-free. A locally nilpotent torsion-free group (cf. Group without torsion) has the uniqueness-of-roots property: If, for elements $a$ and $b$ and any integer $n\ne0$, one has $a^n = b^n$, then $a=b$. Every locally nilpotent torsion-free group $G$ has a Mal'tsev completion, that is, it can be imbedded in a unique locally nilpotent torsion-free group $G^*$ such that all equations of the form $x^n=g$ are solvable in $G^*$, where $n \ne 0$ and $g$ is any element of $G$. This completion is functorial, that is, any homomorphism $f : G_1 \rightarrow G_2$ of locally nilpotent torsion-free groups $G_1$ and $G_2$ can be uniquely extended to a homomorphism $f^* : G_1^* \rightarrow G_2^*$.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[2] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)
How to Cite This Entry:
Locally nilpotent group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_nilpotent_group&oldid=17058
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article