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GroupTheory

  

HallSubgroup

  

construct a Hall subgroup of a finite soluble group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

HallSubgroup( pi, G )

Parameters

pi

-

a list or set of primes

G

-

a soluble permutation group

Description

• 

Let G be a finite group, and let pi be a set of (positive, rational) primes. A Hall pi-subgroup ##  of G is a maximal pi-subgroup of G where, by a pi-subgroup, we mean a subgroup whose order is a pi-number (one whose prime divisors all belong to pi). Equivalently, a subgroup H of a finite group G is a Hall-subgroup if its order and index are relatively prime.

• 

If pi consists of a single prime number p, then a Hall pi-subgroup of G is just a Sylow p-subgroup of G.

• 

A finite group G is soluble if, and only if, for each set pi of primes, G has a Hall pi-subgroup. Moreover, any two Hall pi-subgroups of G are conjugate in G, and every pi-subgroup of G is contained in a Hall pisubgroup.

• 

A finite insoluble group may, or may not, have Hall subgroups.

• 

The HallSubgroup( pi, G ) command constructs a Hall pi-subgroup of a finite soluble group G. The group G must be an instance of a permutation group. Apart from a handful of exceptions, the permutation group G must be soluble; otherwise, an exception is raised.

Examples

withGroupTheory:

GDihedralGroup30

GD30

(1)

ifactorGroupOrderG

2235

(2)

HHallSubgroup2,5,G

H1,202,193,184,175,166,157,148,139,1210,1121,3022,2923,2824,2725,26,1,162,173,184,195,206,217,228,239,2410,2511,2612,2713,2814,2915,30,1,19,7,25,132,20,8,26,143,21,9,27,154,22,10,28,165,23,11,29,176,24,12,30,18

(3)

igcdGroupOrderH,IndexH,G

1

(4)

Hall subgroups can only be computed for soluble groups, in general, so the following example cause an exception to be raised.

HallSubgroup2,3,Symm5

Error, (in HallSubgroup) group must be soluble

However, for certain special cases, a Hall subgroup is returned without exception.

HallSubgroup5,Symm5

<a permutation group on 5 letters>

(5)

HallSubgroup&comma;Symm5

(6)

HallSubgroup2&comma;3&comma;5&comma;Symm5

S5

(7)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]

GroupTheory[SylowSubgroup]