36.12 Sylow Subgroups and Hall Subgroups

returns a Sylow p subgroup of the finite group G. This is a p-subgroup of G whose index in G is coprime to p. SylowSubgroup computes Sylow subgroups via the operation SylowSubgroupOp.

gap> g:=SymmetricGroup(4);;
gap> SylowSubgroup(g,2);
Group([ (1,2), (3,4), (1,3)(2,4) ])

With respect to the following GAP functions, please note that by theorems of P. Hall, a group G is solvable if and only if one of the following conditions holds.

1.

For each prime p dividing the order of G, there exists a p-complement (see SylowComplement).

2.

For each set P of primes dividing the order of G, there exists a P-Hall subgroup (see HallSubgroup).

3.

G has a Sylow system (see SylowSystem).

4.

G has a complement system (see ComplementSystem).

returns a p-Sylow complement of the finite group G. This is a subgroup U of order coprime to p such that the index [G:U] is a p-power. At the moment methods exist only if G is solvable and GAP will issue an error if G is not solvable.

gap> SylowComplement(g,3);
Group([(3,4),(1,4)(2,3),(1,3)(2,4)])

computes a P-Hall subgroup for a set P of primes. This is a subgroup the order of which is only divisible by primes in P and whose index is coprime to all primes in P. The function computes Hall subgroups via the operation HallSubgroupOp. At the moment methods exist only if G is solvable and GAP will issue an error if G is not solvable.

gap> h:=SmallGroup(60,10);;
gap> u:=HallSubgroup(h,[2,3]);
Group([ f1, f2, f3 ])
gap> Size(u);
12

A Sylow system of a group G is a set of Sylow subgroups of G such that every pair of Sylow subgroups from this set commutes as subgroups. Sylow systems exist only for solvable groups. The operation returns fail if the group G is not solvable.

gap> h:=SmallGroup(60,10);;
gap> SylowSystem(h);
[ Group( [ f1, f2 ] ), Group( [ f3 ] ), Group( [ f4 ] ) ]
gap> List(last,Size);
[ 4, 3, 5 ]

A complement system of a group G is a set of Hall- p¢-subgroups of G, where p¢ runs through the subsets of prime factors of |G | that omit exactly one prime. Every pair of subgroups from this set commutes as subgroups. Complement systems exist only for solvable groups, therefore ComplementSystem returns fail if the group G is not solvable.

gap> ComplementSystem(h);
[ Group( [ f3, f4 ] ), Group( [ f1, f2, f4 ] ), Group( 
[ f1, f2, f3 ] ) ]
gap> List(last,Size);
[ 15, 20, 12 ]

returns a list containing one Hall- P subgroup for each set P of primes which occur in the order of G. Hall systems exist only for solvable groups. The operation returns fail if the group G is not solvable.

gap> HallSystem(h);
[ Group( [] ), Group( [ f1, f2 ] ), Group( 
[ f1, f2, f3 ] ), Group( [ f1, f2, f3, f4 ] ), Group( 
[ f1, f2, f4 ] ), Group( [ f3 ] ), Group( 
[ f3, f4 ] ), Group( [ f4 ] ) ]
gap> List(last,Size);
[ 1, 4, 12, 60, 20, 3, 15, 5 ]

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