# Attainable subgroup

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.

#### 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) |

**How to Cite This Entry:**

Attainable subgroup.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Attainable_subgroup&oldid=34066