GAP Manual: 7.31. SylowSubgroup

SylowSubgroup( G, p )

SylowSubgroup returns a Sylow-p-subgroup of the finite group G for a prime p.

Let p be a prime and G be a finite group of order p^n m where m is relative prime to p. Then by Sylow's theorem there exists at least one subgroup S of G of order p^n.

Note that SylowSubgroup sets and tests G.sylowSubgroups[ p ].

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> SylowSubgroup( s4, 2 );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (3,4), (1,2), (1,3)(2,4) ] )
    gap> SylowSubgroup( s4, 3 );
    Subgroup( Group( (1,2,3,4), (1,2) ), [ (2,3,4) ] ) 

The default function GroupOps.SylowSubgroup computes the set of elements of p power order of G, starts with such an element of maximal order and computes the closure (see Closure) with normalizing elements of p power order until a Sylow group is found.