Doubly transitive group action: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A group action on a set is said to be doubly transitive | A group action on a set is said to be '''doubly transitive''' or '''2-transitive''' if the induced action on the set of ordered tuples of distinct elements, is transitive. In other words, given any two ordered tuples each having a pair of distinct elements, there is a group element taking one ordered tuple to the other. | ||
Another way of saying this is that the stabilizers of any two points must intersect in a subgroup whose index in each is 1 less than the index of the subgroups. | Another way of saying this is that the stabilizers of any two points must intersect in a subgroup whose index in each is 1 less than the index of the subgroups. | ||
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===Definition with symbols=== | ===Definition with symbols=== | ||
A group action of a group <math>G</math> on a set <math>S</math> is said to be doubly transitive if given any <math>(x,y)</math> and (x',y')</math> with <math>x \ne y, x' \ne y'</math>, all elements of <math>S</math>, there exists <math>g \in G</math> such that <math>g.x = x'</math> and <math>g.y=y'</math>. | A group action of a group <math>G</math> on a set <math>S</math> is said to be doubly transitive if given any <math>(x,y)</math> and <math>(x',y')</math> with <math>x \ne y, x' \ne y'</math>, all elements of <math>S</math>, there exists <math>g \in G</math> such that <math>g.x = x'</math> and <math>g.y=y'</math>. | ||
Equivalently, if <math>H</math> and <math>K</math> are the stabilizers of two distinct points, and their index is <math>n</math> (which is also the size of <math>S</math>) then <math>H \cap K</math> has index <math>n(n-1)</math>. | Equivalently, it is doubly transitive if it is a [[transitive group action]] and if <math>H</math> and <math>K</math> are the stabilizers of two distinct points, and their index is <math>n</math> (which is also the size of <math>S</math>, by transitivity) then <math>H \cap K</math> has index <math>n(n-1)</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[ | * [[Weaker than::Triply transitive group action]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Doubly set-transitive group action]] | * [[Stronger than::Doubly set-transitive group action]] | ||
* [[Primitive group action]] | * [[Stronger than::Primitive group action]] | ||
* [[Generously transitive group action]] | * [[Stronger than::Generously transitive group action]] | ||
* [[Transitive group action]] | * [[Stronger than::Transitive group action]] | ||
===Related group properties=== | ===Related group properties=== | ||
* [[Doubly transitive group]] is a group possessing a doubly transitive action | * [[Doubly transitive group]] is a group possessing a doubly transitive action | ||
Latest revision as of 00:48, 1 January 2009
This article defines a group action property or a property of group actions: a property that can be evaluated for a group acting on a set.
View a complete list of group action properties|Get help on group action property lookup|Get exploration suggestions
VIEW RELATED: group action property implications | group action property non-implications | {{{context space}}} metaproperty satisfactions | group action metaproperty dissatisfactions | group action property satisfactions |group action property dissatisfactions
Definition
Symbol-free definition
A group action on a set is said to be doubly transitive or 2-transitive if the induced action on the set of ordered tuples of distinct elements, is transitive. In other words, given any two ordered tuples each having a pair of distinct elements, there is a group element taking one ordered tuple to the other.
Another way of saying this is that the stabilizers of any two points must intersect in a subgroup whose index in each is 1 less than the index of the subgroups.
Definition with symbols
A group action of a group on a set is said to be doubly transitive if given any and with , all elements of , there exists such that and .
Equivalently, it is doubly transitive if it is a transitive group action and if and are the stabilizers of two distinct points, and their index is (which is also the size of , by transitivity) then has index .
Relation with other properties
Stronger properties
Weaker properties
- Doubly set-transitive group action
- Primitive group action
- Generously transitive group action
- Transitive group action
Related group properties
- Doubly transitive group is a group possessing a doubly transitive action