[GAP Forum] Fwd: Number of subgroups

[Murthy Sandeep]

To find the (number of) normal subgroups of a group G you use
the Filtered command to filter the list of subgroups of G which are
normal in G.  So if G = D16 is as defined and subs is the list of
subgroups of G as defined in the previous post then the command

gap> Filtered( subs, H -> IsNormal( D16, H ) );

will give you the normal subgroups of D16:

[ Group([  ]), Group([ f4 ]), Group([ f4, f3 ]), Group([ f4, f3, f1 ]), Group([ f4, f3, f2 ]), Group([ f4, f3, f1*f2 ]),
  Group([ f4, f3, f1, f2 ]) ]

Sandeep


> Begin forwarded message:
> 
> From: Murthy Sandeep <sandeep at sandeepmurthy.is>
> Subject: Re: [GAP Forum] Number of subgroups
> Date: 19 April 2015 14:23:40 BST
> To: abdulhakeem alayiwola <lovepgroups at gmail.com>
> Cc: GAP Forum <forum at gap-system.org>
> 
> 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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