Difference between revisions of "Attainable subgroup"
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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 | A subgroup $H$ that can be included in a finite [[normal series]] of a group $G$, i.e. in a series | ||
| − | $$\{1\}\subset H=H_0\subset H_1\subset\ | + | $$\{1\}\subset H=H_0\subset H_1\subset\dotsb\subset H_n=G$$ |
in which each subgroup $H_i$ is a normal subgroup in $H_{i+1}$. The property of a subgroup to be attainable is transitive. An intersection of attainable subgroups is an attainable subgroup. The subgroup generated by two attainable subgroups need not be an attainable subgroup. A group $G$ all subgroups of which are attainable satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. [[Normalizer of a subset]]). Such a group is therefore [[Locally nilpotent group|locally nilpotent]]. | in which each subgroup $H_i$ is a normal subgroup in $H_{i+1}$. The property of a subgroup to be attainable is transitive. An intersection of attainable subgroups is an attainable subgroup. The subgroup generated by two attainable subgroups need not be an attainable subgroup. A group $G$ all subgroups of which are attainable satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. [[Normalizer of a subset]]). Such a group is therefore [[Locally nilpotent group|locally nilpotent]]. | ||
Revision as of 13:37, 14 February 2020
2020 Mathematics Subject Classification: Primary: 20D35 [MSN][ZBL]
A subgroup $H$ that can be included in a finite normal series of a group $G$, i.e. in a series
$$\{1\}\subset H=H_0\subset H_1\subset\dotsb\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) |
Attainable subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_subgroup&oldid=44628