Directed power graph-equivalent groups: Difference between revisions
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{{group eqrel}} | |||
==Definition== | ==Definition== | ||
Two [[group]]s <math>G</math> and <math>H</math> are '''directed power graph-equivalent groups''' if the [[defining ingredient::directed power graph of a group|directed power graph]] of <math>G</math> is isomorphic (as a directed graph) to the [[directed power graph of a group|directed power graph]] of <math>H</math>. | Two [[group]]s <math>G</math> and <math>H</math> are '''directed power graph-equivalent groups''' if the [[defining ingredient::directed power graph of a group|directed power graph]] of <math>G</math> is isomorphic (as a directed graph) to the [[directed power graph of a group|directed power graph]] of <math>H</math>. | ||
==Finite version== | |||
If either of the groups <math>G</math> or <math>H</math> is a [[finite group]], so is the other group, and in this case they are both [[1-isomorphic finite groups]]. | |||
==Facts== | |||
* [[Finite groups are 1-isomorphic iff their directed power graphs are isomorphic]] | |||
* [[Directed power graph-equivalent not implies 1-isomorphic for infinite groups]] | |||
* [[Undirected power graph determines directed power graph for finite group]] | |||
==Relation with other relations== | |||
===Stronger relations=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::isomorphic groups]] || || || || {{intermediate notions short|directed power graph-equivalent groups|isomorphic groups}} | |||
|- | |||
| [[Weaker than::1-isomorphic groups]] || || [[1-isomorphic implies directed power graph-equivalent]] || [[directed power graph-equivalent not implies 1-isomorphic]] || {{intermediate notions short|directed power graph-equivalent groups|1-isomorphic groups}} | |||
|} | |||
===Weaker relations=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::undirected power graph-equivalent groups]] || || || || | |||
|} | |||
Latest revision as of 14:32, 7 August 2010
This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.
Definition
Two groups and are directed power graph-equivalent groups if the directed power graph of is isomorphic (as a directed graph) to the directed power graph of .
Finite version
If either of the groups or is a finite group, so is the other group, and in this case they are both 1-isomorphic finite groups.
Facts
- Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
- Directed power graph-equivalent not implies 1-isomorphic for infinite groups
- Undirected power graph determines directed power graph for finite group
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 | |||
| 1-isomorphic groups | 1-isomorphic implies directed power graph-equivalent | directed power graph-equivalent not implies 1-isomorphic | |FULL LIST, MORE INFO |
Weaker relations
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| undirected power graph-equivalent groups |