# Normalizer condition

From Encyclopedia of Mathematics

*for subgroups*

The condition on a group that every proper subgroup is strictly contained in its normalizer (cf. Normalizer of a subset). Every group satisfying the normalizer condition is a locally nilpotent group. On the other hand, all nilpotent groups, and even groups having an ascending central series ($ZA$-groups), satisfy the normalizer condition. However, there are groups with the normalizer condition and with a trivial centre. Thus, the class of groups with the normalizer condition strictly lies in between the classes of $ZA$-groups and locally nilpotent groups.

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |

#### Comments

#### References

[a1] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1980) |

**How to Cite This Entry:**

Normalizer condition.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Normalizer_condition&oldid=32826

This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article