Group in which every pronormal subgroup is normal
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Contents
Definition
Symbol-free definition
A group in which every pronormal subgroup is normal is a group with the property that any pronormal subgroup of the group is a normal subgroup.
Formalisms
In terms of the subgroup property collapse operator
This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (pronormal subgroup) satisfies the second property (normal subgroup), and vice versa.
View other group properties obtained in this way
Relation with other properties
Stronger properties
- Nilpotent group
- Group in which every subgroup is subnormal
- Group satisfying normalizer condition
- Locally nilpotent group
Facts
For a finite group, every pronormal subgroup being normal is equivalent to the group being a finite nilpotent group.
References
- Transitivity of normality and pronormal subgroups by L. A. Kurdachenko and I. Ya. Subbotin, Combinatorial group theory, discrete groups, and number theory, Volume 421, Page 201 - 210(Year 2006): ^{}^{More info}