# Should one teach to use equality or isomorphism in particular groups?

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.

• 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.
Mar 2, 2022 at 22:02

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.

• 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? Mar 3, 2022 at 14:22

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.

• "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 $S/\sim$.
– user19584
Mar 2, 2022 at 16:17

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.