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 > _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
