25.67 SylowSystem

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:

A list of all prime divisors of the group order of U.

sylowComplements:

contains a list of Sylow complements for all primes in 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:

contains a list of Sylow subgroups for all primes in 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 ] ) ] ) 

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