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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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