GAP:SylowSubgroup

This article is about a GAP function.

Definition

Function type

The SylowSubgroup function takes as input a group and a number and outputs a group.

Behavior

The behavior as follows:

• If the group is a finite group and the number input is a prime number, the function outputs one of the Sylow subgroups for that prime in the group. Note that if the prime does not divide the order of the group, the trivial group is returned.
• If the number input is not a prime number, GAP returns an error beginning with:

<p> must be a prime called from

• If the group is infinite, GAP returns a NoMethodFound error.

Related functions

• GAP:HallSubgroup: This function works for solvable groups, and returns a Hall subgroup for any set of primes.
• GAP:SylowComplement

Examples of usage

Here is an example involving the computation of a 2-Sylow subgroup of symmetric group of degree six:

gap> G := SymmetricGroup(6);
Sym( [ 1 .. 7 ] )
gap> S := SylowSubgroup(G,2);
Group([ (1,2), (3,4), (1,3)(2,4), (5,6) ])
gap> IdGroup(S);
[ 16, 11 ]
gap> IsomorphismGroups(S,DirectProduct(DihedralGroup(8),CyclicGroup(2)));
[ (1,2), (3,4), (1,3)(2,4), (5,6) ] -> [ f1*f2, f1*f2*f3, f1*f3*f4, f4 ]
gap> N := Normalizer(G,S);
Group([ (5,6), (1,4)(2,3), (3,4), (1,2) ])
gap> N = S;
true
gap> Length(ConjugateSubgroups(G,S));
45
gap> T := SylowSubgroup(G,3);
Group([ (1,2,3), (4,5,6) ])
gap> IdGroup(T);
[ 9, 2 ]
gap> U := SylowSubgroup(G,5);
Group([ (1,2,3,4,5) ])
gap> IdGroup(U);
[ 5, 1 ]