25.66 SylowComplements

SylowComplements( U )

SylowComplements returns a Sylow complement system of U. This system S is represented as a record with at least the components S.primes and S.sylowComplements, additionally there may be a component S.sylowSubgroups (see SylowSystem).

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.

SylowComplements uses HallSubgroup (see HallSubgroup) in order to compute the various Sylow complements of U, if no Sylow system is known for U. If a Sylow system { S_1, ... , S_n } is known, SylowComplements computes the various Hall subgroups H_i using the fact that H_i is the product of all S_j except S_i.

SylowComplements sets and checks U.sylowSystem.

    gap> SylowComplements( s4 );
    rec(
      primes := [ 2, 3 ],
      sylowComplements :=
       [ Subgroup( s4, [ b ] ), Subgroup( s4, [ a, c, d ] ) ] ) 

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