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Jul 3, 2020 at 15:44 comment added Oscar Cunningham I think the construction in the accepted answer there also meets your criteria. The universal construction which maps each group to a cogroup object is a 'relatively natural construction within category theory', and two groups have the same order if and only if their corresponding cogroup objects are isomorphic.
Jul 3, 2020 at 15:07 comment added user160570 I should have said 'defined by a relatively natural construction within category theory", not defined in the sense of say first order logic, which is presumably your meaning.
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Jul 3, 2020 at 11:08 comment added Oscar Cunningham 'Order', and everything else, can be defined within the category of groups: mathoverflow.net/questions/352298/…
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S Jul 3, 2020 at 10:32 history answered user160570 CC BY-SA 4.0
S Jul 3, 2020 at 10:32 history made wiki Post Made Community Wiki by user160570