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It is isomorphic to $\mathbb{Z}_{2^8},$ only difference is the symbols usually identifying the elements of the set are from $\{-128, \ldots, 127 \}$ and not $\{0, \ldots, 256\}.$

What is an elegant way to say this in math?

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    I heard the phrase *symmetric representation*, specifically, in the Maple command `mods` which represents $\mathbb{Z}_m$ using $\{-\lfloor \frac{m - 1}{2} \rfloor, \ldots, +\lfloor \frac{m}{2} \rfloor\}$ rather than $\{0, 1, \ldots, m-1 \}.$2012-03-24

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A group needs to satisfy the group axioms, so you need to define the group operation. Presumably you are using addition modulo 256. There is a natural correspondence between the two sets that takes $n \in [-128,127]$ to $\begin {cases} n+256 & n \lt 0 \\n & n \ge 0 \end {cases}$ and you can show that the group structure is maintained.

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    I think we're all in agreement that it is a group and that it's strongly related to $Z_{2^8}$. The question seems to be what to call this group.2012-03-23