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Chris Wensley asked about an "IsDirectSummand" function, and Joachim

Neubueser called attention to the Complementclasses function as a way to

deal with the question.

If I understand correctly, Wensley would like to test whether a given

normal subgroup H is a direct factor of G, and if it is, to find at least

one direct complement. For small groups, checking the complements to H to

see to see if any (or several) are normal is perhaps feasible. For somewhat

larger groups, say of order prime^small, one may have a large number of

nonnormal complements to contend with. In such a case it may be consider-

ably faster to use Complementclasses ( Centralizer (G,H) , Centre (H) ),

which produces a list of all direct complements to H in G by looking in

the smaller group Centralizer (G,H). To get just one complement, replace

Complementclasses by Complement.

If only the existence of a direct complement is required, it seems that

one has to weigh the cost of computing the Centralizer and Center against

the likelihood of hitting a normal complement fairly quickly in a list of

all classes of complements to H.

Checking whether a given normal subgroup is a direct factor is straight-

forward. The question of determining whether G itself is a direct product

appears to be considerably more difficult.

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_--_|\ C.R.B. Wright / \ Charles.Wright@maths.anu.edu.au \_.--._/ wright@math.uoregon.edu v

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