Essentially Unique

An object is essentially unique if its underlying structure is unique, even though its form may vary in ways that do not affect the mathematical content. An object is unique if there is no other object satisfying its defining properties. 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.