GroupTheory
IsCaminaGroup
determine whether a group is a Camina group
Calling Sequence
Parameters
Description
Examples
Compatibility
IsCaminaGroup( G )
G
-
a permutation group
A non-abelian group G is a Camina group if it is not perfect and if, for each g∈G∖H we have gG=g·H, where H is the derived subgroup of G. That is, the conjugacy class of each element of G not in the derived subgroup is equal to its coset of the derived subgroup. (In any group G, the conjugacy class gG of any element g is contained in the coset gH of the derived subgroup.)
Examples of Camina groups are some (but not all) Frobenius groups and extraspecial p-groups, for prime numbers p.
The IsCaminaGroup( G ) command determines, for a permutation group G, whether G is a Camina group. It returns true if G is a Camina group, and returns false otherwise.
withGroupTheory:
IsCaminaGroupQuaternionGroup
true
IsExtraspecialQuaternionGroup
IsCaminaGroupQuaternionGroup4
false
IsExtraspecialQuaternionGroup4
IsCaminaGroupDihedralGroup4
IsCaminaGroupDihedralGroup32
IsCaminaGroupSmallGroup72,41
IsCaminaGroupFrobeniusGroup968,2
IsCaminaGroupFrobeniusGroup300,1
G≔SmallGroup300,23:
IsCaminaGroupG
IsFrobeniusGroupG
H≔DerivedSubgroupG:
cc≔removeg↦ginH,mapRepresentative,ConjugacyClassesG:
nopscc
4
nopsremoveg↦ElementsConjugacyClassg,G=ElementsLeftCosetg,H,cc
2
The GroupTheory[IsCaminaGroup] command was introduced in Maple 2022.
For more information on Maple 2022 changes, see Updates in Maple 2022.
See Also
GroupTheory[IsFrobeniusGroup]
GroupTheory[IsPGroup]
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