1-isomorphism of groups

Definition

Suppose $G_{1}$ and $G_{2}$ are groups. A 1-isomorphism between $G_{1}$ and $G_{2}$ is a bijective 1-homomorphism of groups from $G_{1}$ to $G_{2}$ . In other words, it is a bijective mapping $φ : G_{1} \to G_{2}$ such that for any $g \in G_{1}$ and any integer $n$ , we have:

$φ (g^{n}) = φ (g)^{n}$

If there exists a 1-isomorphism of groups from $G_{1}$ to $G_{2}$ , we say that $G_{1}$ and $G_{2}$ are 1-isomorphic groups.