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Characteristic not implies powering-invariant in nilpotent group

Statement

It is possible to have a nilpotent group G and a characteristic subgroup H of G such that H is not a powering-invariant subgroup of G. In other words, there exists a prime number p such that every element of G has a unique p^{th} root, but there are elements of H whose p^{th} roots are outside H.

Related facts

Proof

See the example in the references.

References