I am wondering the following. Suppose we have some particular space $X$ and $x_1,x_2,x_3,x_4\in X$ has the law of composition that works like Klein four-group. Is it correct to say that the structure that $x_i$'s forms is the Klein four-group or is the structure isomorphic to Klein four-group? I'm confused as when I learned algebra, there was not specified that in what space does the elements of some group, like Klein four-group, belongs to.
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3$\begingroup$ Whether or not it is the Klein 4-group, depends on what you consider to be the Klein 4-group. I'd argue that it is more useful to ask if it is useful to regard it as isomorphic to the Klein 4-group and if so, what choice of isomorphism will you take. After all, this group has non-identity self isomorphisms and so you have some choice in how you match up the generators. Further, different isomorphic copies of the group will come with different representations, etc. $\endgroup$– AdamMar 2, 2022 at 22:02
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3 Answers
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It depends on what foundation of mathematics is being used. Certain foundations, such as Zermelo-Frankel set theory or univalent type theory, have a notion of equality between set-level algebraic structures in a universe, while other foundations, such as the Elementary Theory of the Category of Sets, do not have such a notion of equality. So one may speak of either equality or isomorphism of set-level algebraic structures in the former foundations, but only of isomorphism of set-level algebraic structures in the latter.
Since it is not true that one can speak of equality of set-level algebraic structures in all foundations, it would be better and clearer to simply use isomorphic/isomorphism in teaching for all set-level algebraic structures such as groups, since teaching is foundation-agnostic in practice.
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1$\begingroup$ I'm not familiar the the EToCoS at all, but are you saying that within this structure, one cannot even say the OP's example is equal to itself? $\endgroup$ Mar 3, 2022 at 14:22
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Identifying structures with their isomorphism equivalence classes is a very useful perspective, but it can get really confusing if it's someone's primary perspective.
Specifically, when one starts talking about subgroups (e.g. how many subgroups of G are isomorphic to the Klein four group?), one runs into problems: are all subgroups equal, since they're isomorphic? Are subgroups equal if there's an automorphism of G taking one to the other? Neither of these are particularly the question we'd like the answer to.
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3$\begingroup$ "are all subgroups equal, since they're isomorphic? Are subgroups equal if there's an automorphism of G taking one to the other?" No, because a set or type $S$ with an equivalence relation $\sim$ is not the same as the quotient set $S/\sim$. $\endgroup$– user19584Mar 2, 2022 at 16:17
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One should teach that in some situations you care about strict equality, in some situations you care about isomorphism, and doing a lot of fussing about terminology in a situation where you don't really care about the difference won't help you figure out what to do in a situation where you do care about the difference.
