Isomorphism is Equivalence Relation

Theorem
Isomorphism is an equivalence on the set of algebraic structures.

This result applies to all algebraic structures: rings, groups, R-algebraic structures etc.

Proof
To prove a relation is an equivalence, we need to prove it is reflexive, symmetric and transitive.

So, checking in turn each of the critera for equivalence:

Reflexive
From Identity Mapping is an Automorphism, the identity mapping $I_S: S \to S$ is an automorphism, which is an isomorphism from an algebraic structure onto itself.

So $S \cong S$, and isomorphism is seen to be Reflexive.

Symmetric
If $S \cong T$, then $\exists \phi: S \to T$ such that $\phi$ is an isomorphism.

Thus, from Inverse Isomorphism, $\phi^{-1}: T \to S$ is also an isomorphism, and $T \cong S$.

Thus we have shown that $S \cong T \implies T \cong S$, so $\cong$ is symmetric.

Transitive
Let $S_1 \cong S_2$, and $S_2 \cong S_3$.

We can define these two isomorphisms:


 * $\phi: S_1 \to S_2$
 * $\psi: S_2 \to S_3$

From Composition of Isomorphisms, we have that $\psi \circ \phi$ is an isomorphism between $S_1$ and $S_3$.

Thus we have shown that $\cong$ is transitive.

Thus isomorphism is reflexive, symmetric and transitive, and therefore an equivalence.