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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
| + | #REDIRECT [[Subnormal series]] |
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| − | <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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| − | 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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| − | ====References====
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| − | <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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| − | ====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><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Suzuki, "Group theory" , '''2''' , Springer (1986)</TD></TR></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=18484
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article