Definition:Group Isomorphism

Definition

Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ be groups.

Let $\phi: G \to H$ be a (group) homomorphism.

Then $\phi$ is a group isomorphism iff $\phi$ is a bijection.

That is, $\phi$ is a group isomorphism iff $\phi$ is both a monomorphism and an epimorphism.

If $G$ is isomorphic to $H$, then the notation $G \cong H$ can be used (although notation varies).

Also see

• Results about group isomorphisms can be found here.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 7.1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.5$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 28 \gamma$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 62$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 46, \ \S 47.5 \ \text{(c)}$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Example $2.19$
• John F. Humphreys: A Course in Group Theory (1996): $\S 8$: Definition $8.8$