An object is unique if there is no other object satisfying its defining properties. An object is said to be essentially unique if uniqueness is only referred to the underlying structure, whereas the form may vary in ways that do not affect the mathematical content. For the sake of precision, the decomposition of a positive integer into prime factors is not strictly unique, but rather is essentially unique, because it is unique only up to insignificant formal modifications such as permutations of the factors () or changes of sign (). Similarly, the group of order 2 is essentially unique--despite the evidence that the additive group and the multiplicative group are different--because they are isomorphic groups, which differ only in the names given to their elements and their operations.

# Essentially Unique

## See also

Trivial, Unique, Uniqueness Theorem
*This entry contributed by Margherita
Barile*

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## Cite this as:

Barile, Margherita. "Essentially Unique." From *MathWorld*--A Wolfram Web Resource, created by Eric
W. Weisstein. https://mathworld.wolfram.com/EssentiallyUnique.html