1-isomorphic groups

Definition
Suppose $$G_1$$ and $$G_2$$ are groups. We say that $$G_1$$ and $$G_2$$ are 1-isomorphic if there exists a 1-isomorphism between $$G_1$$ and $$G_2$$, i.e., a 1-homomorphism of groups from $$G_1$$ to $$G_2$$ whose inverse is also a 1-homomorphism. In other words, there is a bijection between $$G_1$$ and $$G_2$$ whose restriction to any cyclic subgroup on either side is an isomorphism to its image.

Historical term
G. A. Miller used the term conformal groups to describe what are referred to here are 1-isomorphic groups. However, the term "conformal group" has a different, much more famous meaning as the group of angle-preserving symmetries of a geometric space.

Finite version
Two finite groups that are 1-isomorphic are termed 1-isomorphic finite groups. There are many equivalent characterizations of 1-isomorphic finite groups.

Facts
Any Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring. This shows that many groups of small prime power order are 1-isomorphic to abelian groups.

Weaker relations
See also 1-isomorphic finite groups.