GroupTheory
IsTransitive
determine whether a permutation group is transitive
Calling Sequence
Parameters
Description
Examples
Compatibility
IsTransitive( G, domain )
G
-
a permutation group
domain
(optional) a stable set (or list) of positive integers
A permutation group G (acting on the set1,2,…,n is transitive if, for any α and β, there is a permutation g in G for which αg=β. Alternatively, G is transitive if it has precisely one orbit.
The domain argument, which is optional and is, by default, equal to the support of G, specifies a stable set under the action of G on which to test the transitivity of G.
The IsTransitive( G ) command returns true if the permutation group G is transitive, and returns false otherwise. The group G must be an instance of a permutation group.
withGroupTheory:
The following group is not transitive because it has two orbits.
G≔PermutationGroupPerm1,2,Perm1,2,3,4,5
G≔1,2,1,2,34,5
IsTransitiveG
false
OrbitsG
11,2,1,2,34,5,41,2,1,2,34,5
However, it is transitive on each of its orbits.
IsTransitiveG,1,2,3
true
IsTransitiveG,4,5
G≔PermutationGroupPerm1,5,Perm1,3,5
G≔1,3,5,1,5
IsTransitiveG,1,2,3,4,5
IsTransitiveAlt10
IsTransitiveFrobeniusGroup12822,2
The GroupTheory[IsTransitive] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[AlternatingGroup]
GroupTheory[FrobeniusGroup]
GroupTheory[Orbit]
GroupTheory[PermutationGroup]
GroupTheory[Transitivity]
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