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Dear GAP-Forum:

Martin Schoenert replied to my email message of 1996/03/22:

Unfortunately it is impossible to recompute the conjugands afterwards.

This is because during the lattice computation the conjugands are

computed as the right transversal of the normalizer of the

representative. But 'RightTransversal' is not guaranteed to return the

same transversal if you call it after the lattice computation.

This surprises me a bit. I thought I could reliably recompute the

SAME right transversal, and so far didn't have any problems with that

assumption. Under what conditions may it happen that subsequent

invocations of RightTransversal yield different results? If different

results can indeed occur (more precisely: different orderings in the

list of cosets), I do not see how GraphicLattice (from XGAP) could

reliably produce the correct result. In the function

GraphicLatticeOps.MakeMaximalSubgroups the right transversals of

subgroup conjugacy class representatives are computed two or more times,

and it seems that it is assumed that always the same result is obtained.

But perhaps I have overlooked some invariance in the topology of the

graph produced by GraphicLattice.

However, it is simple to modify 'MaximalSubgroups' and

'MinimalSupergroups' so that they not only give a subgroup in the form

'[<n>,<m>]' (which means <n>-th conjugacy class, <m>-th subgroup of that

class), but '[<n>,<m>,<c>]' where <c> is a conjugating element.

If you are interested, write me and I will send you the code.

I think it would be a good solution if MaximalSubgroups and

MinimalSupergroups would accept a flag which makes them return

the results in the way you suggest, of even make this the default

behaviour (provided that memory requirements permit, and that nothing

breaks if triples are returned instead of pairs). This would allow

to avoid all these problems.

In any case, I would appreciate receiving your code.

Best regards,

Franz G"ahler

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