In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. In a sense, the existence of such an isomorphism says that the two groups are "the same."
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 $G \cong H$.
Here are a collection of examples:
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 $x \mapsto -2x$.
