# Group in which every pronormal subgroup is normal

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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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

## 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