1-isomorphic finite groups

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


 * 1) They are defining ingredient::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.