Power closed

In mathematics a p-group ${\displaystyle G}$ is called power closed if for every section ${\displaystyle H}$ of ${\displaystyle G}$ the product of ${\displaystyle p^k}$ powers is again a ${\displaystyle p^k}$th power. Regular p-groups are an example of power closed groups. On the other hand, powerful p-groups, for which the product of ${\displaystyle p^k}$ powers is again a ${\displaystyle p^k}$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).

• 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 

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