> < ^ Date: Mon, 14 Dec 1992 16:16:43 +0100
> < ^ From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de >
> ^ Subject: subdirect products

Eric J. Rossin asks in a letter to the GAP-forum for ways of
constructing subdirect products of three 2-groups. I have no complete
answer but just a few comments on his specific questions:

1. I think that working with the polycyclic presentations of 2-groups
as given in the 2-group library of Eamonn O'Brien that is available in
GAP should be most efficient.

2. Subdirect products of *two* factors G and H can best be described
by giving epimorphisms p and q of the two groups onto the same group.
The subdirect product G/\H can then be described as

G/\H  =  { (g.h) |  gp = hq }

This construction is actually used in the GAP command
SubdirectProduct. To classify the subdirect products use actions on
such pairs by the automorphism groups. For a practical use of such
ideas see e.g. a paper by H.Brown, H. Zassenhaus,and myself: On
integral groups I, Num. Math. 19, 1972, p. 386-399, but there may be
better sources, we needed this in the course of looking at reducible
finite unimodular groups which are such subdirect products, but the
aim was not the study of subdirect products.

3. Subdirect products of more than two factors are more difficult to
describe. There are several papers by Robert Remak, in particular
"Ueber Untergruppen direkter Produkte von drei Faktoren", Crelle's J.
fuer die reine u. angewandte Math. 166, 1931 and several others
between 1913 and the latter paper. I do not remember if he gives a
construction that is explicit enough to lend itself to a
classification of such subdirekt products up to isomorphism. To look
at the subgroup lattice of the direct product of the three factors
computed by GAP sounds rather hopeless to me, rather study the work of
Remak first and also have a look into the forward citation index if
you can find later papers taking up Remak's work and quoting him.

Perhaps somebody else in the Gap-forum knows such papers?

Joachim Neubueser

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