# 1-isomorphic finite groups

From Groupprops

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

## Contents

## Definition

Two finite groups are termed **1-isomorphic finite groups** if the following equivalent conditions are satisfied:

- They are 1-isomorphic groups, i.e., there is a bijection between them that restricts to an isomorphism on cyclic subgroups of both sides.
- Their directed power graphs are isomorphic as graphs.
- Their undirected power graphs are isomorphic as graphs.

### Equivalence of definitions

- Equivalence of (1) and (2): finite groups are 1-isomorphic iff their directed power graphs are isomorphic
- Equivalence of (2) and (3): undirected power graph determines directed power graph for finite group

## Facts

- Logarithm map from Lazard Lie group to its Lazard Lie ring is a 1-isomorphism: In particular, this states that a 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 finite groups |

### Weaker relations

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

order-cum-power statistics-equivalent finite groups | the number of elements of any given order and that are any given power (i.e., the order-cum-power statistics) is the same for both groups | 1-isomorphic implies order-cum-power statistics-equivalent | order-cum-power statistics-equivalent not implies 1-isomorphic | |FULL LIST, MORE INFO |

power statistics-equivalent finite groups | the number of elements that are powers for a given is the same | (via order-cum-power statistics-equivalent) | (via order-cum-power statistics-equivalent) | |FULL LIST, MORE INFO |

order statistics-equivalent finite groups | the number of elements of order for each is the same | (via order-cum-power statistics-equivalent) | (via order-cum-power statistics-equivalent) | |FULL LIST, MORE INFO |