GroupTheory
SylowSubgroup
construct a Sylow subgroup of a group
Calling Sequence
Parameters
Description
Examples
Compatibility
SylowSubgroup( p, G )
p
-
a positive rational prime
G
a permutation group or Cayley table group
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 or a Cayley table group.
Note that, if p is not a divisor of the order of G, then the trivial subgroup of G is returned.
withGroupTheory:
G≔SL2,5:
ifactorGroupOrderG
2335
P2≔SylowSubgroup2,G
P2≔1,2,3,45,20,15,106,22,18,147,23,19,118,24,16,129,21,17,13,1,5,3,152,10,4,206,8,18,167,13,19,219,23,17,1112,14,24,22
GroupOrderP2
8
GroupOrderSylowSubgroup3,G
3
GroupOrderSylowSubgroup5,G
5
GroupOrderSylowSubgroup7,G
1
G≔CayleyTableGroupSymm4
G≔ < a Cayley table group with 24 elements >
P≔SylowSubgroup3,G
P≔ < a Cayley table group with 3 elements >
N≔NormaliserP,G
N≔N < a Cayley table group with 24 elements > < a Cayley table group with 3 elements >
Q≔SylowSubgroup2,N
Q≔ < a Cayley table group with 2 elements >
GroupOrderQ
2
The GroupTheory[SylowSubgroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[AlternatingGroup]
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