# Attainable subgroup

2010 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)