GroupTheory
FrattiniSubgroup
construct the Frattini subgroup of a group
Calling Sequence
Parameters
Description
Examples
Compatibility
FrattiniSubgroup( G )
G
-
a group
The Frattini subgroup of a group G is the intersection of the maximal subgroups of G, or G itself in case G has no maximal subgroups.
The Frattini subgroup is equal to 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 S∖g.
The Frattini subgroup of a finite group is nilpotent.
The FrattiniSubgroup( G ) command returns the Frattini subgroup of a group G.
withGroupTheory:
G≔SmallGroup32,5:
F≔FrattiniSubgroupG
F≔Φ1,2,6,11,8,12,7,34,15,18,30,20,31,19,165,10,21,27,23,28,22,149,24,25,32,26,29,13,17,1,42,93,135,176,187,198,2010,1511,2512,2614,1621,2422,2923,3227,3028,31,1,52,103,144,176,217,228,239,1511,2712,2813,1618,2419,2920,3225,3026,31,1,6,8,72,11,12,34,18,20,195,21,23,229,25,26,1310,27,28,1415,30,31,1617,24,32,29,1,82,123,114,205,236,79,2610,2813,2514,2715,3116,3017,3218,1921,2224,29
GroupOrderF
8
IsNilpotentF
true
F≔FrattiniSubgroupDihedralGroup12
F≔ΦD12
2
GroupOrderFrattiniSubgroupAlt4
1
Since a quasicyclic group has no maximal subgroups, it is equal to its Frattini subgroup.
G≔QuasicyclicGroup7
G≔ℤ7∞
FrattiniSubgroupG
ℤ7∞
F≔FrattiniSubgroupDirectProductSemiDihedralGroup6,GL2,3
F≔Φ1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,2,123,234,105,216,87,199,1711,1514,2416,2218,20,27,3028,3129,32,25,30,2926,27,31
AreIsomorphicF,DirectProductFrattiniSubgroupSemiDihedralGroup6,FrattiniSubgroupGL2,3
The GroupTheory[FrattiniSubgroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[AlternatingGroup]
GroupTheory[DihedralGroup]
GroupTheory[GroupOrder]
GroupTheory[IsNilpotent]
GroupTheory[QuasicyclicGroup]
GroupTheory[SmallGroup]
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