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GroupTheory

  

FrattiniSubgroup

  

construct the Frattini subgroup of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

FrattiniSubgroup( G )

Parameters

G

-

a permutation group

Description

• 

The Frattini subgroup of a finite group G is the set of "non-generators" of G.  An element g of G is a non-generator if, whenever G is generated by a set S containing g, it is also generated by Sg.

• 

The Frattini subgroup of G is also equal to the intersection of the maximal subgroups of G. The Frattini subgroup of a finite group is nilpotent.

• 

The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G. The group G must be an instance of a permutation group.

Examples

withGroupTheory:

GSmallGroup32,5:

FFrattiniSubgroupG

FΦ1,2,6,11,8,12,7,34,15,18,30,20,31,19,165,10,21,27,23,28,22,149,24,25,32,26,29,13,17,1,42,93,135,176,187,198,2010,1511,2512,2614,1621,2422,2923,3227,3028,31,1,52,103,144,176,217,228,239,1511,2712,2813,1618,2419,2920,3225,3026,31,1,6,8,72,11,12,34,18,20,195,21,23,229,25,26,1310,27,28,1415,30,31,1617,24,32,29,1,82,123,114,205,236,79,2610,2813,2514,2715,3116,3017,3218,1921,2224,29

(1)

GroupOrderF

8

(2)

IsNilpotentF

true

(3)

FFrattiniSubgroupDihedralGroup12

FΦD12

(4)

GroupOrderF

2

(5)

GroupOrderFrattiniSubgroupAlt4

1

(6)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[DihedralGroup]

GroupTheory[GroupOrder]

GroupTheory[IsNilpotent]

GroupTheory[SmallGroup]