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2 Nilpotent quotients of finitely presented \(Z\)-algebras
 2.1 Computing nilpotent quotients.
 2.2 Computing augmentation quotients of groups.

2 Nilpotent quotients of finitely presented \(Z\)-algebras

2.1 Computing nilpotent quotients.

The following function allows to compute nilpotent quotients for finitely presented associative \(Z\)-algebras given by a number of generators and relators.

2.1-1 NilpotentQuotientFpZAlgebra
‣ NilpotentQuotientFpZAlgebra( A, c )( operation )

Given a finitely presented associative \(Z\)-algebra, this function computes the class-c quotient.

2.2 Computing augmentation quotients of groups.

Let \(I(G)\) denote the augmentation ideal of a group \(G\), then the following functions calculate the class-\(c\) quotient of \(I(G)\) for finitely presented groups and pcp-groups. One can further choose to print the additive structure of the augmentation quotients \(I(G)^i/I(G)^{i+1}\) during computation.

2.2-1 AugmentationQuotientFpGroup
‣ AugmentationQuotientFpGroup( G, c, print )( operation )

Given a finitely presented group G, this function computes the class-c quotient of the augmentation ideal in the integral group ring \(Z\)G. If print is set to true, then the augmentation quotients are printed during computation.

2.2-2 AugmentationQuotientPcpGroup
‣ AugmentationQuotientPcpGroup( G, c, print )( operation )

Given a pcp-group G, this function computes the class-c quotient of the augmentation ideal in the integral group ring \(Z\)G. If print is set to true, then the augmentation quotients are printed during computation.

gap> H := HeisenbergPcpGroup(1);;
gap> AugmentationQuotientPcpGroup(H, 5, true);;
Q_1 = Z^2
Q_2 = Z^4
Q_3 = Z^6
Q_4 = Z^9
Q_5 = Z^12
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