[GAP Forum] isoclinism

[David Joyner]

Sorry for the long delay (final exams, etc). As you probably guessed,
the short answer appears to be that at the moment GAP does not
have the command you want. I personally know nothing about isoclinism
but Joachim Neubueser was kind enough to send me some information which
I'll pass on to you:

"In the case of the isoclinism  question it should not be too difficult
to provide  at least a simple  minded such function  which first tests
isomorphism of the centerfactorgroups and  then checks if one of these
induces an isomorphism of the commutator groups."


To provide some background for those (such as myself) who aren't
familiar with the term, Joachim provided the following information:

"Let me just  briefly brief you on isoclinism. The  notion goes back to
Philip Hall's famous 4 papers in Crelle 182 (1940). It is based on the
simple  observation  that the  value  of  a  commutator [a,b]  of  two
elements a and b  of a group G really depends only  on the cosets of a
and b  modulo the center Z(G)  of G. Hence an  isomorphism from G/Z(G)
onto the Centerfactorgroup H/Z(H) of another group H induces a mapping
of the commutatorgroup G' into the commutatorgroup H'. If this is also
an isomorphism, then the pair  of isomorphisms is called an isoclinism
and  G and  H are  called  isoclinic if  such a  pair of  isomorphisms
exists.

For instance  the dihedral group of  order 8 and  the quaternion group
are  isoclinic, but  isoclinic groups  need not  even be  of  the same
order.   Isoclinic groups of  the same  order form  a 'branch'  of the
isoclinism family, those of minimal  order the 'stem'.  Hall used this
idea for  the classification  of p-groups, and  e.g. the  catalogue of
groups of  order 2^n  up to order  64 by  Marshall Hall and  Senior is
based on this idea.

Philip  Hall had  actually gone  further and  had obtained  a  list of
isoclinism families with  stem groups of order 128 -  and I will never
forget that he sent me, who  then was just a very fresh and absolutely
unknown assistent in Kiel a  handcopy of this list, specially made for
me,  when  I  asked  him  if  copies  exist.   When  I  threw  all  my
correspondence  away this  Spring, I  kept only  this and  gave  it to
Bettina, I think  it is a wonderful document that  Philip Hall was not
only an excellent mathematician, but also a really great man.

Of  course the  importance of  the notion  for  p-group classification
became   fairly  obsolete  with   the  Leedham-Green/Newman   idea  of
classification by  p-uniserial space  groups. However there  are close
links to  representation theory  to which Hall  already points  in his
Crelle papers, but which was worked out with details and extensions in
the Aachen  Habilitationsschrift of  my former student  Juergen Tappe.
This got  published jointly  with that of  Rudolf Beyl  in Heidelberg:
Springer Lecture Notes 958  'Group Extensions, Representations and the
Schur Multiplicator' (1982).

If you  want to have a closer  look, I recommend to  start with Hall's
papers, they are gemstones.

I  hope that  somebody  can be  found  who will  give  the question  a
thought, there  are some theoretical  problems about which  one should
think  before implementing: As  far as  I see  an automorphism  of the
centerfactorgroup   need   not    induce   an  automorphism   of   the
commutatorgroup (although  I have no counterexample at  hand), so that
just testing one  isomorhism of G/Z(G) and H/Z(H)  will not be enough,
but one can perhaps work with cosets of the automorphism group of G/Z(G)
modulo the subgroup of automorphisms induced by automorphisms of G. Or
perhaps do even better?"


+++++++++++++++++++++++++++++++++++++++++++++++++++++


Robert Heffernan wrote:
> Hi,
>
> Is there a function in GAP to determine whether or not two groups are
> isoclinic and, if so, to return an isoclinism (or even all
> isoclinisms) between the two groups?
>
> A search of the documentation doesn't bring anything up.  Perhaps
> somebody has coded this up for their own purposes and would be willing
> to share?
>
> thank you,
> Bob
>