Which algebraic statements are preserved by isomorphisms?

Question

I am asking a general question. Suppose two groups $(G, \circ)$ and $(H, \star)$ are isomorphic. Then as is often stated, "algebraic statements" which hold for $(G, \circ)$ also hold for $(H, \star)$.

Question: Exactly what "algebraic statements" are preserved?

I know that the statement, say for $G$, can only contain the the following:

1. "objects:" $G$, $e_G$ and variables
2. "relations:" $=$, $\in$
3. "functions:" $^{-1}$ (inverse), $\circ$

Is there some precise way to characterize the statements, or is the above list all?

Answer 1

If you are familiar with category theory, then there is a fairly straightforward way to get a broad class of isomorphism-invariant properties.

Recall that a category has a class of objects and, for any two objects $A$ and $B$, a set of morphisms $A \to B$ between these objects.

The finitary statements that can be made about a category can be classified as statements in the language of category theory. In the language of category theory, there is a type of objects and for any objects $A, B$, a type of arrows $A \to B$. Statements in the language of category theory are defined inductively as all statements of the form

• $f = g$ where $f, g$ are both arrow terms $A \to B$ for some objects $A, B$
• For any statements in the language of category theory, any combination of these statements using the logical operators $\land, \lor, \neg, \implies, \top, \bot$
• For any statement in the language of category theory $P(f)$ where $f$ has type $A \to B$, the statements $\forall f : A \to B, P(f)$ and $\exists f : A \to B, P(f)$
• For any statement in the language of category theory $P(A)$ where $A$ has object type, the statements $\forall A, P(A)$ and $\forall B, P(B)$

Notice the critical bit is in the propositions of the form $f = g$. We CANNOT compare two objects for equality, and we can ONLY compare two arrows for equality when they have the same type. So we have excluded statements like $G_1 = G_2$ when we are discussing groups $G_1$ and $G_2$.

Now consider the category of sets. Consider a statement $P(G, +)$ in the language of category theory, where $+ : G^2 \to G$.

Given two isomorphic groups $(G_1, +_1), (G_2, +_2)$, one can always prove the statement $P(G_1, +_1) \iff P(G_2, +_2)$.

So any statement that can be stated (1) in the language of the category of sets and (2) only in terms of the group operator, the statement is preserved by isomorphism.

Let's ask ourselves what kinds of statements can be made in the language of category theory.

Can we discuss elements of a set $A$ in the language of category theory? Yes. Let $1$ be a fixed 1-element set. Then elements of $A$ correspond exactly to functions $1 \to A$. Thus, we can compare two elements to equality, apply a function to an element, and quantify over all the elements in a set.

Can we discuss subsets of a set $A$ in the language of category theory? Yes. Rather than discussing the subsets themselves, we can discuss injective functions $f : B \to A$. Note that injective functions into a set $A$ correspond in a canonical way to subsets of $A$.

Can we discuss things like the group having finite order in the language of category theory? Yes. It turns out that we can write "$A$ is finite" in the language of category theory.

Can we discuss things like inverses and the identity element? Yes. The identity element is the unique element $e : 1 \to G$ such that for all elements $x : 1 \to G$, $e + x = x + e = x$. So we can specify the identity element using the language of category theory together with the axioms of a group. We can do the same with inverses.

In fact, the language of category theory is so broad that it's possible to come up with a set theory solely based in the language of category theory that is exactly as powerful as normal set theory. If you take a categorical set theory to be your foundation, then literally every statement you can make about a given group is preserved under isomorphism.