1-isomorphic finite groups: Difference between revisions
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* Equivalence of (1) and (2): [[finite groups are 1-isomorphic iff their directed power graphs are isomorphic]] | * 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]] | * 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== | ==Relation with other relations== | ||
Latest revision as of 22:12, 9 November 2011
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:
- 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 |