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Is there any natural group structure on the set $I_p = \{x \in \mathbb{Z}[1/p] \mid |x| < p/2\}$?

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This is a really strange question without any motivation. What could "natural" possibly mean? –  Qiaochu Yuan Nov 18 '09 at 4:30

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This set doesn't have a subgroup structure as a subset of the reals, but if p is an odd prime, you can certainly lift the additive group law on Qp/pZp to this set. An alternative way to look at it that works for arbitrary odd numbers p is to lift the p-power roots of unity in the complex numbers by the map x -> e2 pi ix/p.

If p is even, you probably have to make more choices.

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