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GroupTheory

  

IsCPGroup

  

determine whether a group is a (CP)-group

  

IsCP1Group

  

determine whether a group is a (CP1)-group

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

IsCPGroup( G )

IsCP1Group( G )

Parameters

G

-

a group

Description

• 

A group G is a (CP)-group if each of its elements has prime-power order, where the prime may depend upon the element. A group of prime-power order is a (CP)-group, but there are (CP)-groups, such as S3 , whose order is divisible by more than one prime. This is equivalent to the condition that the centralizer of each non-trivial element is a p-group, for some prime p depending upon the element. It is also equivalent for finite groups to the Gruenberg-Kegel graph of the group being totally disconnected.

• 

A group G is a (CP1)-group if each of its non-trivial elements has prime order where, again, the prime may depend upon the element. Equivalently, a group is a (CP1)-group if the centralizer of each non-trivial element contains only elements of order dividing p, for a prime p depending upon the element. The symmetric group S3 is again an example of a (CP1)-group not of prime exponent.

• 

It is a consequence of these definitions that every (CP1)-group is a (CP)-group. Both (CP)-groups and (CP1)-groups are (CN)-groups.

• 

The IsCPGroup( G ) command returns true if the group G is a (CP)-group and returns false otherwise.

• 

The IsCP1Group( G ) command returns true if the group G is a (CP1)-group and returns false otherwise.

Examples

withGroupTheory:

The symmetric group S3 furnishes an example of a (CP)-group that is not a group of prime power order, and a (CP1)-group that is not of prime exponent.

IsCPGroupSymm3

true

(1)

IsPGroupSymm3

false

(2)

IsCP1GroupSymm3

true

(3)

ExponentSymm3

6

(4)

The symmetric group S4 is also a (CP)-group, but symmetric groups of larger degree are not. However, S4 is not a (CP1)-group.

IsCPGroupSymm4

true

(5)

IsCP1GroupSymm4

false

(6)

orseqIsCPGroupSymmn,n=5..10

false

(7)

A cyclic group is a (CP)-group if, and only if, it has prime power order.

seqn=IsCPGroupCyclicGroupn,n=1..20

1=true,2=true,3=true,4=true,5=true,6=false,7=true,8=true,9=true,10=false,11=true,12=false,13=true,14=false,15=false,16=true,17=true,18=false,19=true,20=false

(8)

And cyclic groups are (CP1)-groups precisely when the order is a prime.

seqn=IsCP1GroupCyclicGroupn,n=1..20

1=true,2=true,3=true,4=false,5=true,6=false,7=true,8=false,9=false,10=false,11=true,12=false,13=true,14=false,15=false,16=false,17=true,18=false,19=true,20=false

(9)

Dihedral groups are (CP)-groups just when the degree is a prime power.

seqn=IsCPGroupDihedralGroupn,n=1..20

1=true,2=true,3=true,4=true,5=true,6=false,7=true,8=true,9=true,10=false,11=true,12=false,13=true,14=false,15=false,16=true,17=true,18=false,19=true,20=false

(10)

And, dihedral groups of prime degree are the only ones that are (CP1)-groups.

seqn=IsCP1GroupDihedralGroupn,n=1..20

1=true,2=true,3=true,4=false,5=true,6=false,7=true,8=false,9=false,10=false,11=true,12=false,13=true,14=false,15=false,16=false,17=true,18=false,19=true,20=false

(11)

Groups of prime exponent are (CP1)-groups.

GSmallGroup81,12:

ExponentG

3

(12)

IsCP1GroupG

true

(13)

GSmallGroup98,4

G1,23,84,75,106,911,2212,2413,2314,1915,2116,2017,2618,2527,4428,4629,4530,4831,4732,3933,4134,4035,4336,4237,5038,4951,7052,6953,7254,7155,7456,7357,6458,6359,6660,6561,6862,6775,8876,8777,9078,8979,8480,8381,8682,8591,9892,9793,9694,95,1,3,11,27,32,14,42,7,19,39,44,22,85,12,28,51,57,33,156,13,29,52,58,34,169,20,40,63,69,45,2310,21,41,64,70,46,2417,30,53,75,79,59,3518,31,54,76,80,60,3625,42,65,83,87,71,4726,43,66,84,88,72,4837,55,77,91,93,81,6138,56,78,92,94,82,6249,67,85,95,97,89,7350,68,86,96,98,90,74,1,5,17,37,38,18,62,9,25,49,50,26,103,12,30,55,56,31,134,15,35,61,62,36,167,20,42,67,68,43,218,23,47,73,74,48,2411,28,53,77,78,54,2914,33,59,81,82,60,3419,40,65,85,86,66,4122,45,71,89,90,72,4627,51,75,91,92,76,5232,57,79,93,94,80,5839,63,83,95,96,84,6444,69,87,97,98,88,70

(14)

IsCPGroupG

true

(15)

IsCP1GroupG

true

(16)

GSmallGroup72,41

G < a permutation group on 72 letters with 5 generators >

(17)

IsCPGroupG

true

(18)

IsCP1GroupG

false

(19)

GSmallGroup192&comma;182

G < a permutation group on 192 letters with 7 generators >

(20)

IsCPGroupG

false

(21)

IndexPCore2&comma;G&comma;G

6

(22)

The group B222 is the Frobenius group of order 20 and is a (CP)-group.

IsCPGroupSuzuki2B22

true

(23)

Other Frobenius groups provide important examples of (CP)-groups.

IsCPGroupFrobeniusGroup72&comma;2

true

(24)

These are all the simple (CP)-groups (M. Suzuki).

IsCPGroupAlt5

true

(25)

IsCPGroupSuzuki2B28

true

(26)

IsCPGroupSuzuki2B232

true

(27)

IsCPGroupPSL2&comma;7

true

(28)

IsCPGroupPSL2&comma;8

true

(29)

IsCPGroupPSL2&comma;9

true

(30)

IsCPGroupPSL2&comma;17

true

(31)

IsCPGroupPSL3&comma;4

true

(32)

GraphTheory:-DrawGraphGruenbergKegelGraphPSL3&comma;4

The only simple (in fact, the only insoluble) (CP1)-group is the alternating group A5 .

IsCP1GroupAlt5

true

(33)

An example of an infinite (CP)-group.

GQuasicyclicGroup17

G17

(34)

IsCPGroupG

true

(35)

GroupOrderG

(36)

See Also

GraphTheory[DrawGraph]

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[CyclicGroup]

GroupTheory[DihedralGroup]

GroupTheory[Exponent]

GroupTheory[GroupOrder]

GroupTheory[GruenbergKegelGraph]

GroupTheory[Index]

GroupTheory[IsCNGroup]

GroupTheory[PCore]

GroupTheory[ProjectiveSpecialLinearGroup]

GroupTheory[QuasicyclicGroup]

GroupTheory[SmallGroup]

GroupTheory[Suzuki2B2]

GroupTheory[SymmetricGroup]

seq

with