GroupTheory/EARNS - Maple Help
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GroupTheory

  

EARNS

  

compute an elementary abelian regular normal subgroup of a primitive permutation group

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

EARNS( G )

Parameters

G

-

PermutationGroup; a permutation group

Description

• 

For a permutation group G, an "EARNS" is a normal subgroup of G that is elementary abelian and acts regularly on the domain of action of G. A permutation group may, or may not, possess an EARNS.

• 

The EARNS( G ) command returns an EARNS for a permutation group G, provided that one exists, and returns FAIL if G has no EARNS.

• 

It is clear that for a permutation group to posess an EARNS it must be transitive and its support must have prime power cardinality. Therefore, EARNS returns FAIL if either of these conditions is not true.

• 

In general, for Maple to identify an EARNS for a permutation group the group must either be primitive or a Frobenius group (or both). If G is neither primitive nor a Frobenius group, then EARNS may raise an exception indicating that the group is imprimitive and that Maple cannot, in that case, determine whether or not G has an EARNS.

Examples

withGroupTheory:

GSymm3

GS3

(1)

EARNSG

1,2,3

(2)

EARNSAlt4

1,42,3,1,23,4

(3)

GSymm4

GS4

(4)

EEARNSG

E1,42,3,1,23,4

(5)

IsNormalE,G

true

(6)

IsRegularE

true

(7)

IsElementaryE

true

(8)

A group acting on a set not of prime power cardinality can have no EARNS.

GCyclicGroup10

GC10

(9)

EARNSG

FAIL

(10)

IsTransitiveG

true

(11)

typeSupportLengthG,'primepower'

false

(12)

An intransitive group cannot posess an EARNS.

GGroupPerm1,2,3,4,5

G1,2,34,5

(13)

IsTransitiveG

false

(14)

EARNSG

FAIL

(15)

GGroupPerm2,7,4,8,6,5,3,Perm2,4,3,6,8,7,Perm1,2,3,4,5,6,7,8

G2,7,4,8,6,5,3,2,4,36,8,7,1,23,45,67,8

(16)

EEARNSG

E1,42,35,86,7,1,72,83,54,6,1,23,45,67,8

(17)

Primitive Frobenius groups always have an EARNS, the Frobenius kernel.

GFrobeniusGroup14520,2

G < a permutation group on 121 letters with 5 generators >

(18)

IsPrimitiveG

true

(19)

EEARNSG&colon;

AreIsomorphicE&comma;ElementaryGroup11&comma;2

true

(20)

IsSubgroupE&comma;FrobeniusKernelGandIsSubgroupFrobeniusKernelG&comma;E

true

(21)

GSymm2048

GS2048

(22)

EARNSG

FAIL

(23)

IsPrimitiveG

true

(24)

GAlt55

GA3125

(25)

EARNSG

FAIL

(26)

IsPrimitiveG

true

(27)

A regular elementary abelian transitive group is its own EARNS, even if it does not act primitively.

GTransitiveGroup27&comma;4&colon;

EARNSG

1&comma;2&comma;34&comma;5&comma;67&comma;8&comma;910&comma;11&comma;1213&comma;14&comma;1516&comma;17&comma;1819&comma;20&comma;2122&comma;23&comma;2425&comma;26&comma;27&comma;1&comma;6&comma;262&comma;4&comma;273&comma;5&comma;257&comma;10&comma;138&comma;11&comma;149&comma;12&comma;1516&comma;21&comma;2317&comma;19&comma;2418&comma;20&comma;22&comma;1&comma;11&comma;192&comma;12&comma;203&comma;10&comma;214&comma;15&comma;225&comma;13&comma;236&comma;14&comma;247&comma;16&comma;258&comma;17&comma;269&comma;18&comma;27

(28)

IsPrimitiveG

false

(29)

IsRegularG

true

(30)

Some imprimitive Frobenius groups have an EARNS.

GTransitiveGroup25&comma;9

G2&comma;3&comma;5&comma;46&comma;14&comma;24&comma;177&comma;11&comma;23&comma;208&comma;13&comma;22&comma;189&comma;15&comma;21&comma;1610&comma;12&comma;25&comma;19&comma;1&comma;7&comma;11&comma;20&comma;232&comma;8&comma;12&comma;16&comma;243&comma;9&comma;13&comma;17&comma;254&comma;10&comma;14&comma;18&comma;215&comma;6&comma;15&comma;19&comma;22&comma;1&comma;25&comma;19&comma;12&comma;102&comma;21&comma;20&comma;13&comma;63&comma;22&comma;16&comma;14&comma;74&comma;23&comma;17&comma;15&comma;85&comma;24&comma;18&comma;11&comma;9

(31)

EARNSG

Fitt2&comma;3&comma;5&comma;46&comma;14&comma;24&comma;177&comma;11&comma;23&comma;208&comma;13&comma;22&comma;189&comma;15&comma;21&comma;1610&comma;12&comma;25&comma;19&comma;1&comma;7&comma;11&comma;20&comma;232&comma;8&comma;12&comma;16&comma;243&comma;9&comma;13&comma;17&comma;254&comma;10&comma;14&comma;18&comma;215&comma;6&comma;15&comma;19&comma;22&comma;1&comma;25&comma;19&comma;12&comma;102&comma;21&comma;20&comma;13&comma;63&comma;22&comma;16&comma;14&comma;74&comma;23&comma;17&comma;15&comma;85&comma;24&comma;18&comma;11&comma;9

(32)

IsPrimitiveG

false

(33)

But not all do.

GTransitiveGroup16&comma;63

G2&comma;3&comma;45&comma;15&comma;96&comma;13&comma;127&comma;14&comma;108&comma;16&comma;11&comma;1&comma;5&comma;3&comma;72&comma;6&comma;4&comma;89&comma;14&comma;11&comma;1610&comma;13&comma;12&comma;15

(34)

IsPrimitiveG

false

(35)

IsFrobeniusPermGroupG

true

(36)

EARNSG

FAIL

(37)

In most cases, however, an exception is raised if the input to EARNS is imprimitive.

GWreathProductCyclicGroup3&comma;CyclicGroup3

G1&comma;2&comma;3&comma;1&comma;4&comma;72&comma;5&comma;83&comma;6&comma;9

(38)

IsPrimitiveG

false

(39)

EARNSG

Error, (in GroupTheory:-EARNS) group must be primitive

See Also

GroupTheory

GroupTheory[IsElementary]

GroupTheory[IsNormal]

GroupTheory[IsPrimitive]

GroupTheory[IsRegular]