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 $\varphi :G_{1}\to G_{2}$ such that for any $g\in G_{1}$ and any integer $n$, we have:

$\!\varphi (g^{n})=\varphi (g)^{n}$

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.