Dear GAP Forum members,

Chad Lower had asked:

I am doing research on groups. Specifically,

=E2=94=82Bn=E2=94=82 is the order of the group Bn. Bn is the collection =
of all subgroups of B with order n.
Our goal is to find all groups so that each =E2=94=82Bn=E2=94=82 will div=
ide the order of the group. We are looking at finite groups only.

For example, B =3D Z2 X Z3 has order 6.
There is 1 subgroup of order 2 : {(1,0)} -- 1 divides 6
There is 1 subgroup of order 3 : {(0,1)} -- 1 divides 6
There is 1 subgroup of order 6 : {(1,1)} -- 1 divides 6
so Z2 X Z3 has this property.

But Z2 X Z2 does not work.  Z2 X Z2 has order 4.
There are 3 subgroups of order 3 : {(1,0), (0,1), (1,1)} -- 3 does NOT di=
vide 4

I have general cases for:

Zp X Zp X ... XZp where p is prime
Zp^n X Zp^n where p is prime and n is a natural number
Zp1p2 X Zp1p2 where p1 and p2 are distinct primes

I think I have general cases for others, but the sets are often too big t=
o do by hand and I was wondering if GAP had any way to find the possible =
subgroups and tell me their order so I can try to verify some of my other=
cases. If you need more clarification, feel free to ask. Thank you in =
advance for any help you are able to provide.

Chad T. Lower
a.k.a. Chopper
http://chadtlower.tripod.com/

There are three aspects of this letter to be answered:

First: Please do not use exotic font encoding in letters to the
Forum. I, likely for one, have not been able to decode it and also
some younger colleagues who are pretty good at these things, have not
been able to do it in reasonable time. Please rather use plain text or
at best some TEX.

If I understand right what you say without using the encoded formulae,
then you want to study groups in which the number of subgroups of a
certain order divides the group order and for this you ask how you can
use GAP to find the number of subgroups of a given order in a given
finite group.

So secondly let me answer this last question.

The function

ConjugacyClassesSubgroups(g)

will give you a list of all conjugacy classes of subgroups of the
group g.

You can the use the various functions for lists to get all information
about subgroups. E.g. to get the number of subgroups of order 4 in the
symmetric group S_4 type:

g := SymmetricGroup(4);
c := ConjugacyClassesSubgroups(g);
f := Sum(Filtered(c, cl -> Size(cl[1]) = 4), x -> Size(x));

and you will get the result that there are 7 subgrups of order 4 in S_4.

For the many possibilities of using list functions in GAP read the
manual chapter on lists. Note in the above that Size(cl) will give you
the size of the class, i.e. the number of conjugates, but Size(cl[1])
will give you the size, i.e. order of the first group in the class!

Finally, I am afraid that the problem you want to investigate does not
look very promising to me. The condition that the number of subgroups
of given order should divide the group order looks rather unnatural to
me.

There are on the other hand some papers on the number of subgroups of
given order, mainly using the idea of the Eulerian functions of a
group, that might interest you. See for instance:

G.E. Wall
Some applications of the Eulerian Functions of a Finite Group.
Journal Austral. Math. Soc. 2 (1961) 35 - 59.

as well as papers quoted therein and also possibly papers that quote
this one and that you may find via the Science Quotation Index.

With kind regards Joachim Neubueser