Difference between revisions of "Normalizer condition"
From Encyclopedia of Mathematics
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| − | The condition on a [[Group|group]] that every proper subgroup is strictly contained in its normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]). Every group satisfying the normalizer condition is a [[Locally nilpotent group|locally nilpotent group]]. On the other hand, all nilpotent groups, and even groups having an ascending central series ( | + | The condition on a [[Group|group]] that every proper subgroup is strictly contained in its normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]). Every group satisfying the normalizer condition is a [[Locally nilpotent group|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. |
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Latest revision as of 15:27, 10 August 2014
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
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