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.
GAP 3.4.4