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.
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.
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