Composite of Group Isomorphisms is Isomorphism

Theorem

Let $\struct {G_1, \circ}$, $\struct {G_2, *}$ and $\struct {G_3, \oplus}$ be groups.

Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ and $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$ be group isomorphisms.

Then the composite of $\psi$ with $\phi$ is also a group isomorphism.

Proof

A group isomorphism is a group homomorphism which is also a bijection.

From Composite of Group Homomorphisms is Homomorphism, $\psi \circ \phi$ is a group homomorphism.

From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.2$. Some lemmas on homomorphisms: Lemma $\text{(iv)}$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Factoring Morphisms