SylowSystem( U )
SylowSystem
returns a Sylow system { S_1, ... , S_n } of an ag
group U. The system S is represented as a record with at least the
components S.primes
and S.sylowSubgroups
, additionally there may
be a component S.sylowComplements
, see SylowComplements for
information about this addtional component.
primes
:
sylowComplements
:S.primes
, so that if the i.th element of
S.primes
is p, then the i.th element of
sylowComplements
is a Sylow-p-complement of U.
sylowSubgroups
:S.primes
, such that if the i.th element of
S.primes
is p, then the i.th element of
S.sylowSubgroups
is a Sylow-p-subgroup of U.
A Sylow system of a group U is a system of Sylow subgroups S_i for each prime divisor of the group order of U such that S_i * S_j = S_j * S_i is fulfilled for each pair i,j.
SylowSystem
uses SylowComplements
(see SylowSystem) in order to
compute the various Sylow complements H_i of U. Then the Sylow
system is constructed using the fact that the intersection S_i of all
Sylow complements H_j except H_i is a Sylow subgroup and that all
these subgroups S_i form a Sylow system of U. See Gla87.
SylowSystem
sets and checks S.sylowSystem
.
gap> z5 := CyclicGroup( AgWords, 5 ); Group( c5 ) gap> D := DirectProduct( z5, s4 ); Group( a, b1, b2, b3, b4 ) gap> D.name := "z5Xs4";; gap> SylowSystem( D ); rec( primes := [ 2, 3, 5 ], sylowComplements := [ Subgroup( z5Xs4, [ a, b2 ] ), Subgroup( z5Xs4, [ a, b1, b3, b4 ] ), Subgroup( z5Xs4, [ b1, b2, b3, b4 ] ) ], sylowSubgroups := [ Subgroup( z5Xs4, [ b1, b3, b4 ] ), Subgroup( z5Xs4, [ b2 ] ), Subgroup( z5Xs4, [ a ] ) ] )
GAP 3.4.4