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text: '[summary(brief):A phrase mathematicians use when saying "we only care about the [structure_mathematics structure] of an [object_mathematics object], not about specific implementation details of the object".]\n\n[summary: "The property $P$ holds up to isomorphism" is a phrase which means "we might say an object $X$ has property $P$, but that's an abuse of notation. When we say that, we really mean that there is an object [4f4 isomorphic] to $X$ which has property $P$". Essentially, it means "the property might not hold as stated, but if we replace the idea of *equality* by the idea of *isomorphism*, then the property holds".\n\nRelatedly, "The object $X$ is [5ss well-defined] up to isomorphism" means "if we replace $X$ by an object isomorphic to $X$, we still obtain something which satisfies the definition of $X$."]\n\n"The property $P$ holds up to isomorphism" is a phrase which means "we might say an object $X$ has property $P$, but that's an abuse of notation. When we say that, we really mean that there is an object [4f4 isomorphic] to $X$ which has property $P$". Essentially, it means "the property might not hold as stated, but if we replace the idea of *equality* by the idea of *isomorphism*, then the property holds".\n\nRelatedly, "The object $X$ is [5ss well-defined] up to isomorphism" means "if we replace $X$ by an object isomorphic to $X$, we still obtain something which satisfies the definition of $X$."\n\n# Examples\n\n## Groups of order $2$\n\nThere is only one [-3gd] of [3gg order] $2$ *up to isomorphism*.\nWe can define the object "group of order $2$" as "the group with two elements"; this object is well-defined up to isomorphism, in that while there are several different groups of order $2$ %%note: Two such groups are $\\{0,1\\}$ with the operation "addition [5ns modulo] $2$", and $\\{e, x \\}$ with [-54p] $e$ and the operation $x^2 = e$.%%, any two such groups are isomorphic.\nIf we don't think of isomorphic objects as being "different", then there is only one distinct group of order $2$.',
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