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GroupTheory

  

FittingSubgroup

  

construct the Fitting subgroup of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

FittingSubgroup( G )

Parameters

G

-

a permutation group

Description

• 

The Fitting subgroup of a finite group G is the unique largest normal nilpotent subgroup of G. Its existence and uniqueness is guaranteed by Fitting's Theorem, which asserts that the product of a family of normal and nilpotent subgroups of a finite group G is again a normal and nilpotent subgroup of G.

• 

The Fitting subgroup of G is also equal to the (direct) product of the p-cores of G, as p ranges over the prime divisors of the order of G.

• 

If G is a soluble group, then the Fitting subgroup of G is nontrivial.

• 

The FittingSubgroup( G ) command constructs the Fitting subgroup  of a group G. The group G must be an instance of a permutation group.

Examples

withGroupTheory:

GGroupPerm1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,Perm1,2,5,3,4,7,6,9,8,11,10,13,16,15,12,14

G1,23,45,67,89,1011,1213,1415,16,1,2,5,34,76,98,1110,13,16,1512,14

(1)

FFittingSubgroupG

FFitt1,23,45,67,89,1011,1213,1415,16,1,2,5,34,76,98,1110,13,16,1512,14

(2)

GroupOrderF

16

(3)

FFittingSubgroupAlt4

FFittA4

(4)

GroupOrderF

4

(5)

GroupOrderFittingSubgroupAlt6

1

(6)

Compatibility

• 

The GroupTheory[FittingSubgroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[PCore]

GroupTheory[PermutationGroup]