This section describes some functions for creating finite dimensional K[G]-modules for a finitely presented group G. These are obtained from the conjugation action of G on a basis for a maximal elementary abelian quotient of a normal subgroup H of G, where H has finite index in G. For a complete description of the functions available for working with K[G]-modules we refer to Chapter MODULES OVER AN ALGEBRA AND GROUP REPRESENTATIONS.
Note that the function GModuleAction can be used to extract the matrix representation associated to an K[G]-module.
All operations described in this subsection may require a closed coset table for at least one subgroup of an fp-group. If a closed coset table is needed and has not been computed, a coset enumeration will be invoked. If the coset enumeration does not produce a closed coset table, a runtime error is reported.
Experienced users can control the behaviour of such indirectly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function SetGlobalTCParameters. For a detailed description of the available parameters and their meanings, we refer to Subsec. Interactive Coset Enumeration.
Let G be a finitely presented group and A a normal subgroup of G of finite index. Given any prime p, the maximal p-elementary abelian quotient of A can be viewed as a GF(p)[G]-module Mp. This function determines all primes p such that Mp is not trivial (i.e. zero-dimensional) and the dimensions of the corresponding modules Mp. The return value is a multiset S. If 0∉S, the maximal abelian quotient of A is finite and the multiplicity of p is the dimension of Mp. If S contains 0 with multiplicity m, the maximal abelian quotient of A contains m copies of Z. In this case, Mp is non-trivial for every prime p. The rank of Mp in this case is the sum of m and the multiplicity of p in S.
Let G be a finitely presented group, A a normal subgroup of finite index in G and B a normal subgroup of G contained in A. Given any prime p, the maximal p-elementary abelian quotient of A/B can be viewed as a GF(p)[G]-module Mp. This function determines all primes p such that Mp is not trivial (i.e. zero-dimensional) and the dimensions of the corresponding modules Mp. The return value is a multiset S. If 0∉S, the maximal abelian quotient of A/B is finite and the multiplicity of p is the dimension of Mp. If S contains 0 with multiplicity m, the maximal abelian quotient of A/B contains m copies of Z. In this case, Mp is non-trivial for every prime p. The rank of Mp in this case is the sum of m and the multiplicity of p in S.
Given a finitely presented group G, a normal subgroup A of finite index in G and a prime p, create the GF(p)[G]-module M corresponding to the conjugation action of G on the maximal p-elementary abelian quotient of A. The function also returns the epimorphism π: A -> M.Note that normality of A in G is not checked. The results for invalid input data are undefined.
Given a finitely presented group G, a normal subgroup A of G of finite index, a normal subgroup B of G contained in A and a prime p, create the GF(p)[G]-module M corresponding to the conjugation action of G on the maximal p-elementary abelian quotient of A/B.The integer p can be omitted, if the maximal elementary abelian quotient of A/B is a p-group for some prime p. Note, however, that the computation is much faster, if a prime is specified.
The function also returns the epimorphism π: A -> M.
Note that normality of A and B in G is not checked. The results for invalid input data are undefined.
Given a map f:A -> M from a normal subgroup A of an fp-group G onto a GF(p)[G]-module M and a submodule N of M, try to compute the preimage of N under f using a fast pullback method. If successful, the preimage is returned as subgroup of A.If the pullback works, it is in general faster than a direct computation of the preimage using the preimage operator and it produces a more concise generating set for the preimage; see the following example. In cases where the pullback fails, a runtime error is reported and a preimage construction should be used instead.
Consider the group G defined by the presentation
< a,b,c,d,e | a^4, b^42, c^6, e^3, b^a=b^-1, [a,c], [a,d], [a,e],
c^b=c*e, d^b=d^-1, e^b=e^2, d^c=d*e, e^c=e^2, [d,e] >.
> F<a,b,c,d,e> := FreeGroup(5); > G<a,b,c,d,e> := quo< F | a^4, b^42, c^6, e^3, > b^a=b^-1, (a,c), (a,d), (a,e), > c^b=c*e, d^b=d^-1, e^b=e^2, > d^c=d*e, e^c=e^2, > (d,e) >;The finite index subgroup H of G generated by c, d, e is normal in G.
> H := sub< G | c,d,e >; > Index(G, H); 168 > IsNormal(G, H); trueWe check, for which characteristics the action of G on H yields non-trivial modules.
> GModulePrimes(G, H);
{* 0, 2, 3 *}
We construct the (F)3[G]-module M given by the action of G on the
maximal 3-elementary abelian quotient of H and the natural epimorphism
π from H onto the additive group of M.
> M, pi := GModule(G, H, 3); > M; GModule M of dimension 2 over GF(3)Using the function Submodules, we obtain the submodules of M. Their preimages under π are precisely the normal subgroups of G which are contained in H and contain (ker)(π).
> submod := Submodules(M); > time nsgs := [ m @@ pi : m in submod ]; Time: 11.640 > [ Index(G, s) : s in nsgs ]; [ 1512, 504, 504, 168 ]The generating sets for the normal subgroups obtained in this way, contain in general many redundant generators. (E.g. each will contain a generating set for (ker)(π).)
> [ NumberOfGenerators(s) : s in nsgs ]; [ 19, 20, 20, 21 ]Optimised generating sets can be obtained using the function ReduceGenerators.
> nsgs_red := [ ReduceGenerators(s) : s in nsgs ]; > [ NumberOfGenerators(s) : s in nsgs_red ]; [ 2, 2, 3, 2 ]Alternatively, and in fact this is the recommended way, we can use the function Pullback to compute the preimages of the submodules under π. Note that the generating sets for the preimages computed this way contain fewer redundant generators.
> time nsgs := [ Pullback(pi, m) : m in submod ]; Time: 8.560 > [ Index(G, s) : s in nsgs ]; [ 1512, 504, 504, 168 ] > [ NumberOfGenerators(s) : s in nsgs ]; [ 4, 4, 3, 2 ]
< a, b, c, d, e |
a^5, b^5, c^6, d^5, e^3, b^a = bd,\cr
(a,c), (a,d), (a,e), (b,c), (b,d), (b,e), (c,d), (c,e), (d,e) >.
> G<a,b,c,d,e> := FPGroup< a,b,c,d,e | > a^5, b^5, c^6, d^5, e^3, b^a = b*d, > (a,c), (a,d), (a,e), (b,c), (b,d), (b,e), > (c,d), (c,e), (d,e) >;Obviously the subgroup of G generated by b, c, d, e is normal in G.
> H := sub< G | b,c,d,e >; > IsNormal(G, H); trueWe use the function GModulePrimes to determine the set of primes p for which the action of G on the maximal p-elementary abelian quotient of H induces a nontrivial GF(p)[G]-module.
> P := GModulePrimes(G, H); > 0 in P; false0 is not contained in P, i.e. the maximal free abelian quotient of H is trivial. Hence, there are only finitely many primes, satisfying the condition above.
We loop over the distinct elements of P and for each element p we construct the induced GF(p)[G]-module, print its dimension and check whether it is decomposable. Note that the dimension of the module for p must be equal to the multiplicity of p in P.
> for p in MultisetToSet(P) do > M := GModule(G, H, p); > dim := Dimension(M); > decomp := IsDecomposable(M); > > assert dim eq Multiplicity(P, p); > > print "prime", p, ": module of dimension", dim; > if decomp then > print " has a nontrivial decomposition"; > else > print " is indecomposable"; > end if; > end for; prime 2 : module of dimension 1 is indecomposable prime 3 : module of dimension 2 has a nontrivial decomposition prime 5 : module of dimension 2 is indecomposable