1-isomorphic groups

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This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on 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. Further information: Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring

Relation with other relations

Stronger relations

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorphic groups				|FULL LIST, MORE INFO

Weaker relations

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
directed power graph-equivalent groups				|FULL LIST, MORE INFO
undirected power graph-equivalent groups				|FULL LIST, MORE INFO

See also 1-isomorphic finite groups#Weaker properties.