# 1-isomorphic groups

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

## Definition

Suppose and are groups. We say that and are **1-isomorphic** if there exists a 1-isomorphism between and , i.e., a 1-homomorphism of groups from to whose inverse is also a 1-homomorphism. In other words, there is a bijection between and 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 | Directed power graph-equivalent groups|FULL LIST, MORE INFO |