Power closed

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In mathematics a p-group G {\displaystyle G} {\displaystyle G} is called power closed if for every section H {\displaystyle H} {\displaystyle H} of G {\displaystyle G} {\displaystyle G} the product of p k {\displaystyle p^{k}} {\displaystyle p^{k}} powers is again a p k {\displaystyle p^{k}} {\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 p k {\displaystyle p^{k}} {\displaystyle p^{k}} powers is again a p k {\displaystyle p^{k}} {\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, Th. 16).

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