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 defining ingredient::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.