GroupTheory
FittingSubgroup
construct the Fitting subgroup of a group
Calling Sequence
Parameters
Description
Examples
Compatibility
FittingSubgroup( G )
G
-
a permutation group
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.
withGroupTheory:
G≔GroupPerm1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,Perm1,2,5,3,4,7,6,9,8,11,10,13,16,15,12,14
G≔1,23,45,67,89,1011,1213,1415,16,1,2,5,34,76,98,1110,13,16,1512,14
F≔FittingSubgroupG
F≔Fitt1,23,45,67,89,1011,1213,1415,16,1,2,5,34,76,98,1110,13,16,1512,14
GroupOrderF
16
F≔FittingSubgroupAlt4
F≔FittA4
4
GroupOrderFittingSubgroupAlt6
1
The GroupTheory[FittingSubgroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[AlternatingGroup]
GroupTheory[PCore]
GroupTheory[PermutationGroup]
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