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GroupTheory

  

HallSystem

  

compute a Hall system for a finite soluble group

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

HallSystem( G )

Parameters

G

-

a finite soluble group

Description

• 

Let G be a finite soluble group.  A Hall system for G is a collection C of Hall π-subgroups of G, one for each subset π of the prime divisors of the order of G. Note that this includes both G itself, as well as the trivial subgroup of G.

• 

A Hall system for G exists provided that G is a soluble group, and conversely.

• 

The HallSystem( G ) command constructs a Hall system for the soluble group G. If the group G is not soluble, then an exception is raised.

Examples

withGroupTheory:

CHallSystemSymm3

C1,2,1,2,3,1,2,3,1,3,

(1)

mapGroupOrder,C

1,2,3,6

(2)

CHallSystemAlt4

C1,2,3,2,3,4,1,4,2,1,23,4,1,32,4,

(3)

mapGroupOrder,C

1,3,4,12

(4)

CHallSystemWreathProductSymm3,CyclicGroup2

C1,2,1,2,3,1,42,53,6,4,6,5,1,3,24,6,5,1,42,53,6,1,6,3,42,5,1,34,6,

(5)

mapGroupOrder,C

1,8,9,72

(6)

GFrobeniusGroup42,1

G2,73,64,5,2,3,54,7,6,1,2,3,4,5,6,7

(7)

ifactorGroupOrderG

237

(8)

CHallSystemG

C2,73,64,5,2,3,54,7,6,1,2,3,4,5,6,7,1,4,23,5,6,1,2,3,4,5,6,7,1,62,53,4,1,2,3,4,5,6,7,1,72,63,5,1,6,52,3,7,1,2,3,4,5,6,7,1,6,52,3,7,1,72,63,5,

(9)

mapGroupOrder,C

1,2,3,6,7,14,21,42

(10)

GDirectProductDihedralGroup15,Symm3

G < a permutation group on 18 letters with 4 generators >

(11)

ifactorGroupOrderG

22325

(12)

CHallSystemG

C < a permutation group on 18 letters with 4 generators > &comma;1&comma;4&comma;7&comma;10&comma;132&comma;5&comma;8&comma;11&comma;143&comma;6&comma;9&comma;12&comma;15&comma;1&comma;6&comma;112&comma;7&comma;123&comma;8&comma;134&comma;9&comma;145&comma;10&comma;15&comma;16&comma;17&comma;18&comma;1&comma;4&comma;7&comma;10&comma;132&comma;5&comma;8&comma;11&comma;143&comma;6&comma;9&comma;12&comma;15&comma;16&comma;17&comma;2&comma;153&comma;144&comma;135&comma;126&comma;117&comma;108&comma;9&comma;&comma; < a permutation group on 18 letters with 4 generators > &comma;1&comma;4&comma;7&comma;10&comma;132&comma;5&comma;8&comma;11&comma;143&comma;6&comma;9&comma;12&comma;15&comma;1&comma;6&comma;112&comma;7&comma;123&comma;8&comma;134&comma;9&comma;145&comma;10&comma;15&comma;16&comma;17&comma;18&comma;1&comma;132&comma;123&comma;114&comma;105&comma;96&comma;814&comma;15&comma;16&comma;17

(13)

mapGroupOrder&comma;C

1&comma;4&comma;5&comma;9&comma;20&comma;36&comma;45&comma;180

(14)

Since GL2&comma;4 is an insoluble group, attempting to compute a Hall system for this group causes an exception to be raised.

HallSystemGL2&comma;4

Error, (in GroupTheory:-HallSystem) group must be soluble

IsSolubleGL2&comma;4

false

(15)

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[CyclicGroup]

GroupTheory[DihedralGroup]

GroupTheory[DirectProduct]

GroupTheory[FrobeniusGroup]

GroupTheory[GeneralLinearGroup]

GroupTheory[GroupOrder]

GroupTheory[IsSoluble]

GroupTheory[SymmetricGroup]

GroupTheory[WreathProduct]

ifactor

map

with