# Isomorphism is Equivalence Relation

## Theorem

Isomorphism is an equivalence on a set of magmas.

This result applies to all magmas: 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 criteria for equivalence:

### Reflexive

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

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

$\Box$

### Symmetric

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

Thus, from Inverse of Algebraic Structure Isomorphism is 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.

$\Box$

### 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 Composite of Isomorphisms in Algebraic Structure is Isomorphism, we have that $\psi \circ \phi$ is an isomorphism between $S_1$ and $S_3$.

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

$\Box$

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

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 7.3$. Isomorphism - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Theorem $26.1$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{H}$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 62$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 46.1$ Isomorphic groups - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Proposition $8.9$