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Difference between revisions of "Attainable subgroup"

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A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013910/a0139101.png" /> that can be included in a finite normal series of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013910/a0139102.png" />, i.e. in a series
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A subgroup $H$ that can be included in a finite normal series of a group $G$, i.e. in a series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013910/a0139103.png" /></td> </tr></table>
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$$\{1\}\subset H=H_0\subset H_1\subset\ldots\subset H_n=G$$
  
in which each subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013910/a0139104.png" /> is a normal subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013910/a0139105.png" />. The property of a subgroup to be attainable is transitive. An intersection of attainable subgroups is an attainable subgroup. The subgroup generated by two attainable subgroups need not be an attainable subgroup. A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013910/a0139106.png" /> all subgroups of which are attainable satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. [[Normalizer of a subset|Normalizer of a subset]]). Such a group is therefore locally nilpotent.
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in which each subgroup $H_i$ is a normal subgroup in $H_{i+1}$. The property of a subgroup to be attainable is transitive. An intersection of attainable subgroups is an attainable subgroup. The subgroup generated by two attainable subgroups need not be an attainable subgroup. A group $G$ all subgroups of which are attainable satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. [[Normalizer of a subset|Normalizer of a subset]]). Such a group is therefore locally nilpotent.
  
 
====References====
 
====References====

Revision as of 07:13, 15 July 2014

A subgroup $H$ that can be included in a finite normal series of a group $G$, i.e. in a series

$$\{1\}\subset H=H_0\subset H_1\subset\ldots\subset H_n=G$$

in which each subgroup $H_i$ is a normal subgroup in $H_{i+1}$. The property of a subgroup to be attainable is transitive. An intersection of attainable subgroups is an attainable subgroup. The subgroup generated by two attainable subgroups need not be an attainable subgroup. A group $G$ all subgroups of which are attainable satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. Normalizer of a subset). Such a group is therefore locally nilpotent.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)


Comments

Instead of attainable subgroup, the term accessible subgroup is used in [1]. In the Western literature the term subnormal subgroup is standard for this kind of subgroup.

References

[a1] M. Suzuki, "Group theory" , 2 , Springer (1986)
How to Cite This Entry:
Attainable subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_subgroup&oldid=32436
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article