Normalizer condition

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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.


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



[a1] D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)
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Normalizer condition. Encyclopedia of Mathematics. URL:
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