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| − | {{TEX|done}}{{MSC|20D35}}
| + | #REDIRECT [[Subnormal 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
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| − | $$\{1\}\subset H=H_0\subset H_1\subset\dotsb\subset H_n=G$$
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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]].
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| − | ====References====
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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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| − | ====Comments====
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| − | 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.
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| − | ====References====
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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>
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| − | </table>
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Latest revision as of 09:55, 3 January 2021
How to Cite This Entry:
Attainable subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_subgroup&oldid=44628
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article