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Difference between revisions of "Attainable subgroup"

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#REDIRECT [[Subnormal series]]
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|Normalizer of a subset]]). Such a group is therefore locally nilpotent.
 
 
 
====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></table>
 
 
 
 
 
 
 
====Comments====
 
Instead of attainable subgroup, the term accessible subgroup is used in [[#References|[1]]]. In the Western literature the term subnormal subgroup is standard for this kind of subgroup.
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''2''' , Springer  (1986)</TD></TR></table>
 

Latest revision as of 09:55, 3 January 2021

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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