[GAP Forum] Fwd: Number of subgroups

[Murthy Sandeep]

> On 19 Apr 2015, at 06:10, abdulhakeem alayiwola <lovepgroups at gmail.com> wrote:
> 
> how many central automorphism does D-16 have?
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The command

gap> Order( Centre( AutomorphismGroup( DihedralGroup( 16 ) ) ) );
2

Sandeep

> Begin forwarded message:
> 
> From: Murthy Sandeep <sandeep at sandeepmurthy.is>
> Subject: Fwd: [GAP Forum] Number of subgroups
> Date: 19 April 2015 15:08:32 BST
> To: abdulhakeem alayiwola <lovepgroups at gmail.com>
> Cc: GAP Forum <forum at gap-system.org>
> 
> 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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>>> Forum mailing list
>>> Forum at mail.gap-system.org
>>> http://mail.gap-system.org/mailman/listinfo/forum
>> 
>