A group isomorphism $$\phi\colon G \to H$$ is a bijective group homomorphism. Alternatively you could say a homomorphism $$\phi\colon G \to H$$ is an isomorphism if there exists another homomorphism $$\phi^{-1}\colon H \to G$$ such that $$\phi^{-1}\phi$$ is the identity on $$G$$ and $$\phi\phi^{-1}$$ is the identity on $$H$$. If such an isomorphism $$\phi\colon G \to H$$ exists, we say that $$G$$ and $$H$$ are isomorphic, which means that they are structurally identical as groups. This is usually signified by writing $$G \cong H$$.
• The groups $$(\mathbb{R},+)$$, the real numbers equipped with addition, and $$(\mathbb{R}^{+},\times)$$, the positive real numbers equipped with multiplication, are isomorphic. The function $$\exp\colon\mathbb{R}\to \mathbb{R}^{+}$$ that sends $$x$$ to $$\mathrm{e}^x$$ is a group isomorphism that demonstrates this.
• The group of integers $$\mathbb{Z}$$ under addition is isomorphic to its subgroup containing the elements $$\{\dotsc, -2, -1, 0, 1, 2, \dotsc\}$$; there are two isomorphisms that demonstrate this: either the function $$x \mapsto 2x$$ or the function $$x \mapsto -2x$$.