59.7 NonsplitExtension

NonsplitExtension(chr, [vec] )

chr must be a cohomology-record, created by a call of CHR(G,p,F,mats), where F is a finitely presented group. If present, vec must be a list of integers of length equal to the dimension over K = GF(p) of the second cohomology group H^2(G,M) of the group G in its action on the module M defined by the matrices mats. NonsplitExtension calculates and returns a presentation of a nonsplit extension of M by G. Since there may be many such extensions, and the equivalence classes of these extensions are in one-one correspondence with the nonzero elements of H^2(G,M), the optional second parameter can be used to specify an element of H^2(G,M) as a vector. The default value of this vector is [1,0,...,0]. The set of generators of the finitely presented group that is returned is a union of two sets, which are in one-one correspondence with the generators of F and of M (as an abelian group), respectively.

The relators fall into three classes:

:

a) Those that say that M is an abelian group of exponent p;
b) Those that define the action of the generators of F on those of M; and
c) Those that give the values of the relators of F as elements of M.

(Note: It is not particularly efficient to call SecondCohomologyDimension first to calculate the dimension of H^2(G,M), which must of course be known if the second parameter is to be given; it is preferable to call NonsplitExtension immediately without the second parameter (which will return one nonsplit extension), and then to call SecondCohomologyDimension, which will at that stage return the required dimension immediately - all subsequent calls of NonsplitExtension on chr will also yield immediate results.)

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