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I'd like to ask can we characterize the structure of finitely generated infinite $p$-group which has a unique subgroup of order $p$?

Can we say that these group are residually nilpotent?

Any comments are welcome.

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    Every finite subgroup of such$a$group should be cyclic. Also this group has a non-trivial center. Have you an example of such a group that is not abelian?2013-11-19

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