normalizer condition

A group is said to satisfy the normalizer condition if every proper subgroupMathworldPlanetmath is properly contained in its own normalizerMathworldPlanetmathPlanetmath. That is, a group G satisfies the normalizer condition if and only if H<NG(H) for all H<G. A group that satisfies the normalizer condition is sometimes called an N-group.

Every nilpotent groupMathworldPlanetmath is an N-group, and every N-group is locally nilpotentPlanetmathPlanetmath. In particular, a finitely generated group is an N-group if and only if it is nilpotent.

A group satisfies the normalizer condition if and only if all its subgroupsMathworldPlanetmathPlanetmath are ascendant.

Title normalizer condition
Canonical name NormalizerCondition
Date of creation 2013-03-22 16:14:41
Last modified on 2013-03-22 16:14:41
Owner yark (2760)
Last modified by yark (2760)
Numerical id 6
Author yark (2760)
Entry type Definition
Classification msc 20F19
Synonym normaliser condition
Related topic LocallyNilpotentGroup
Defines N-group