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GroupTheory

  

SylowSubgroup

  

construct a Sylow subgroup of a group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SylowSubgroup( p, G )

Parameters

p

-

a positive rational prime

G

-

a permutation group

Description

• 

Let G be a finite group, and let p be a positive (rational) prime.  A Sylow p-subgroup of G is a maximal p-subgroup of G where, by a p-subgroup, we mean a subgroup whose order is a power of p. The Sylow theorems assert that, for a prime divisor p of the order of a finite group G, there is a Sylow p-subgroup of G and that all Sylow p-subgroups of G are conjugate in G.  Moreover, the number of Sylow p-subgroups of G is congruent to 1 modulo p.

• 

The SylowSubgroup( p, G ) command constructs a Sylow p-subgroup of a group G. The group G must be an instance of a permutation group.

• 

Note that, if p is not a divisor of the order of G, then the trivial subgroup of G is returned.

Examples

withGroupTheory:

GAlternatingGroup4

GA4

(1)

ifactorGroupOrderG

223

(2)

P2SylowSubgroup2,G

P2<a permutation group on 4 letters>

(3)

GroupOrderP2

4

(4)

GroupOrderSylowSubgroup3&comma;G

3

(5)

GroupOrderSylowSubgroup5&comma;G

1

(6)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[AlternatingGroup]