1-isomorphic finite groups

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

Definition

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

1. They are 1-isomorphic groups, i.e., there is a bijection between them that restricts to an isomorphism on cyclic subgroups of both sides.
2. Their directed power graphs are isomorphic as graphs.
3. 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 $d^{th}$ powers for a given $d$ 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 $d$ for each $d$ is the same	(via order-cum-power statistics-equivalent)	(via order-cum-power statistics-equivalent)	|FULL LIST, MORE INFO