36.11 Specific and Parametrized Subgroups

gap> TrivialSubgroup(g);
Group( () )

If G and H are two groups of elements in the same family, this operation returns the group generated by all commutators [ g, h ] = g^-1 h^-1 g h (see Comm) of elements g Î G and h Î H , that is the group á [ g, h ] \mid g Î G , h Î H ñ.

gap> CommutatorSubgroup(Group((1,2,3),(1,2)),Group((2,3,4),(3,4)));
Group([ (1,4)(2,3), (1,3,4) ])
gap> Size(last);
12

The derived subgroup G¢ of G is the subgroup generated by all commutators of pairs of elements of G. It is normal in G and the factor group G/G¢ is the largest abelian factor group of G.

gap> DerivedSubgroup(g);
Group( [ (1,3,2), (2,4,3) ] )

CommutatorLength returns the minimal number n such that each element in the derived subgroup (see DerivedSubgroup) of the group G can be written as a product of (at most) n commutators of elements in G.

gap> CommutatorLength( g );
1

The Fitting subgroup of a group G is its largest nilpotent normal subgroup.

gap> FittingSubgroup(g);
Group( [ (1,2)(3,4), (1,3)(2,4) ] )

The Frattini subgroup of a group G is the intersection of all maximal subgroups of G.

gap> FrattiniSubgroup(g);
Group( () )

returns a Prefrattini subgroup of the finite solvable group G. A factor M/N of G is called a Frattini factor if M/N £ f(G/N) holds. The group P is a Prefrattini subgroup of G if P covers each Frattini chief factor of G, and if for each maximal subgroup of G there exists a conjugate maximal subgroup, which contains P. In a finite solvable group G the Prefrattini subgroups form a characteristic conjugacy class of subgroups and the intersection of all these subgroups is the Frattini subgroup of G.

gap> G := SmallGroup( 60, 7 );
<pc group of size 60 with 4 generators>
gap> P := PrefrattiniSubgroup(G);
Group([ f2 ])
gap> Size(P);
2
gap> IsNilpotent(P);
true
gap> Core(G,P);
Group([])
gap> FrattiniSubgroup(G);
Group([])

is the smallest normal subgroup of G that has a solvable factor group.

gap> PerfectResiduum(Group((1,2,3,4,5),(1,2)));
Group([ (1,3,2), (2,4,3), (2,3)(4,5) ])

is the radical of G, i.e., the largest solvable normal subgroup of G.

gap> RadicalGroup(SL(2,5));
<group of 2x2 matrices of size 2 in characteristic 5>
gap> Size(last);
2

The socle of the group G is the subgroup generated by all minimal normal subgroups.

gap> Socle(g);
Group([ (1,4)(2,3), (1,2)(3,4) ])

is the supersolvable residuum of the group G, that is, its smallest normal subgroup with supersolvable factor group.

gap> SupersolvableResiduum(g);
Group([ (1,2)(3,4), (1,4)(2,3) ])

The p-rump of a group G is the subgroup G¢G^p for a prime p.

@example missing!@

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