Power closed

In mathematics a p-group [math]\displaystyle{ G }[/math] is called power closed if for every section [math]\displaystyle{ H }[/math] of [math]\displaystyle{ G }[/math] the product of [math]\displaystyle{ p^k }[/math] powers is again a [math]\displaystyle{ p^k }[/math]th power. Regular p-groups are an example of power closed groups. On the other hand, powerful p-groups, for which the product of [math]\displaystyle{ p^k }[/math] powers is again a [math]\displaystyle{ p^k }[/math]th power are not power closed, as this property does not hold for all sections of powerful p-groups.

The power closed 2-groups of exponent at least eight are described in (Mann 2005).

References

• Mann, Avinoam (2005), "The number of generators of finite p-groups", Journal of Group Theory 8 (3): 317–337, doi:10.1515/jgth.2005.8.3.317, ISSN 1433-5883