Twisted subgroup: Difference between revisions
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Note that the second condition is redundant when <math>K</math> is a finite subset of <math>G</math>. Since twisted subgroups are usually studied in the context of finite groups, the condition is typically omitted from the definition. It is, however, necessary for the definition to behave nicely for infinite groups. The corresponding definition without this condition is better called [[twisted submonoid]]. | Note that the second condition is redundant when <math>K</math> is a finite subset of <math>G</math>. Since twisted subgroups are usually studied in the context of finite groups, the condition is typically omitted from the definition. It is, however, necessary for the definition to behave nicely for infinite groups. The corresponding definition without this condition is better called [[twisted submonoid]]. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::subgroup]] || || || || {{intermediate notions short|twisted subgroup|subgroup}} | |||
|- | |||
| [[Weaker than::2-powered twisted subgroup]] || twisted subgroup within which every element has a unique square root. || || || {{intermediate notions short|twisted subgroup|2-powered twisted subgroup}} | |||
|} | |||
===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::1-closed subset]] || nonempty subset, contains the cyclic subgroup generated by any element in it || || || {{intermediate notions short|1-closed subset|twisted subgroup}} | |||
|- | |||
| [[Stronger than::symmetric subset]] || nonempty subset, contains the identity and closed under taking inverses || || || {{intermediate notions short|symmetric subset|twisted subgroup}} | |||
|} | |||
==Property theory== | ==Property theory== | ||
===Associates=== | ===Associates=== | ||
{{further|[[associate of twisted subgroup is twisted subgroup]]}} | |||
Let <math>K</math> be a twisted subgroup of <math>G</math>. Then, for any <math>a</math> in <math>K</math>, the sets <math>Ka</math> and <math>a^{-1}K</math> are equal and form another twisted subgroup. Such a twisted subgroup is termed an '''associate''' of <math>K</math>. The relation of being associate is an equivalence relation and we are interested in studying twisted subgroups upto the equivalence relation of being associates. | Let <math>K</math> be a twisted subgroup of <math>G</math>. Then, for any <math>a</math> in <math>K</math>, the sets <math>Ka</math> and <math>a^{-1}K</math> are equal and form another twisted subgroup. Such a twisted subgroup is termed an '''associate''' of <math>K</math>. The relation of being associate is an equivalence relation and we are interested in studying twisted subgroups upto the equivalence relation of being associates. | ||
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===Intersection=== | ===Intersection=== | ||
See [[intersection of twisted subgroups is twisted subgroup]]. | |||
==References== | ==References== | ||
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==External links== | ==External links== | ||
* [http:// | * [http://paws.wcu.edu/tsfoguel/publications.html Tuval Foguel's list of publications] which include publications on twisted subgroups | ||
Latest revision as of 02:16, 6 September 2013
This article defines a property of subsets of groups
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Definition
Definition with symbols
A subset of a group is termed a twisted subgroup if it satisfies the following two conditions:
- The identity element belongs to
- For every ,
- Given in , the element is in
Note that the second condition is redundant when is a finite subset of . Since twisted subgroups are usually studied in the context of finite groups, the condition is typically omitted from the definition. It is, however, necessary for the definition to behave nicely for infinite groups. The corresponding definition without this condition is better called twisted submonoid.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| subgroup | |FULL LIST, MORE INFO | |||
| 2-powered twisted subgroup | twisted subgroup within which every element has a unique square root. | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| 1-closed subset | nonempty subset, contains the cyclic subgroup generated by any element in it | |FULL LIST, MORE INFO | ||
| symmetric subset | nonempty subset, contains the identity and closed under taking inverses | |FULL LIST, MORE INFO |
Property theory
Associates
Further information: associate of twisted subgroup is twisted subgroup
Let be a twisted subgroup of . Then, for any in , the sets and are equal and form another twisted subgroup. Such a twisted subgroup is termed an associate of . The relation of being associate is an equivalence relation and we are interested in studying twisted subgroups upto the equivalence relation of being associates.
Intersection
See intersection of twisted subgroups is twisted subgroup.
References
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
from Foguel's article
External links
- Tuval Foguel's list of publications which include publications on twisted subgroups