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Difference between revisions of "Attainable subgroup"

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(Category:Group theory and generalizations)
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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
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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
  
 
$$\{1\}\subset H=H_0\subset H_1\subset\ldots\subset H_n=G$$
 
$$\{1\}\subset H=H_0\subset H_1\subset\ldots\subset H_n=G$$
  
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.
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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]]). Such a group is therefore [[Locally nilpotent group|locally nilpotent]].
  
 
====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></table>
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<table>
 
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<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>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''2''' , Springer  (1986)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''2''' , Springer  (1986)</TD></TR>
[[Category:Group theory and generalizations]]
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</table>

Revision as of 18:20, 5 April 2018

2020 Mathematics Subject Classification: Primary: 20D35 [MSN][ZBL]

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=34066
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article