[GAP Forum] Number of subgroups

[Sandeep Murthy]

I assume you mean the dihedral group of order 16, so define it:

gap> D16 := DihedralGroup( 16 );
<pc group of size 16 with 4 generators>

and then run the command

gap> LatticeSubgroups( D16 );

which should display

gap> <subgroup lattice of <pc group of size 16 with 4 generators>, 11 classes, 19 subgroups>

This displays information about the lattice of subgroups of D16 using the conjugacy relation for
subgroups (http://www.gap-system.org/Manuals/doc/ref/chap39.html#X7FA267497CFC0550).
The classes are the equivalence classes, 11 in this case, and there are 19 subgroups in total.

You cannot access the subgroups from the lattice directly (it is not a list of subgroups), but through
the conjugacy classes.  To do this you have to call the ConjugacyClassesOfSubgroups( <lattice> )
function with a given lattice, which gives you a list of the classes, and then flatten that list.  So you
could do something like:

gap> subs := Flat( List( cls, c -> Elements( c ) ) ) );
[ Group([  ]), Group([ f4 ]), Group([ f1 ]), Group([ f1*f3 ]), Group([ f1*f4 ]), Group([ f1*f3*f4 ]), Group([ f1*f2 ]),
 Group([ f1*f2*f3 ]), Group([ f1*f2*f4 ]), Group([ f1*f2*f3*f4 ]), Group([ f4, f3 ]), Group([ f4, f1 ]),
 Group([ f1*f3, f4 ]), Group([ f4, f1*f2 ]), Group([ f1*f2*f3, f4 ]), Group([ f4, f3, f1 ]), Group([ f4, f3, f2 ]),
 Group([ f4, f3, f1*f2 ]), Group([ f4, f3, f1, f2 ]) ]

An alternative to finding the number of subgroups is to load the Sonata package
(http://www.gap-system.org/Packages/sonata.html) using

gap> LoadPackage( “Sonata” );

and then run the command

gap> Number( Subgroups( D16 ) );

which should display 19.

Sandeep


> On 19 Apr 2015, at 11:35, abdulhakeem alayiwola <lovepgroups at gmail.com> wrote:
> 
> how do i get the number of subgroups of D-16?
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Sandeep Murthy
s.murthy at mykolab.com