View source for Attainable subgroup

{{TEX|done}}{{MSC|20D35}}

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 group|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>