GAP (HAP commands) - Chapter 21: FpG-modules

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21 FpG-modules
21.1
21.1-1 CompositionSeriesOfFpGModules
21.1-2 DirectSumOfFpGModules
21.1-3 FpGModule
21.1-4 FpGModuleDualBasis
21.1-5 FpGModuleHomomorphism
21.1-6 DesuspensionFpGModule
21.1-7 RadicalOfFpGModule
21.1-8 RadicalSeriesOfFpGModule
21.1-9 GeneratorsOfFpGModule
21.1-10 ImageOfFpGModuleHomomorphism
21.1-11 GroupAlgebraAsFpGModule
21.1-12 IntersectionOfFpGModules
21.1-13 IsFpGModuleHomomorphismData
21.1-14 MaximalSubmoduleOfFpGModule
21.1-15 MaximalSubmodulesOfFpGModule
21.1-16 MultipleOfFpGModule
21.1-17 ProjectedFpGModule
21.1-18 RandomHomomorphismOfFpGModules
21.1-19 Rank
21.1-20 SumOfFpGModules
21.1-21 SumOp
21.1-22 VectorsToFpGModuleWords

### 21 FpG-modules

#### 21.1

##### 21.1-1 CompositionSeriesOfFpGModules

`‣ CompositionSeriesOfFpGModules` | ( global variable )

Inputs an FpG-module M and returns a list of FpG-modules that constitute a composition series for M.

Examples:

##### 21.1-2 DirectSumOfFpGModules

`‣ DirectSumOfFpGModules`( M, N ) | ( function )

`‣ DirectSumOfFpGModules`( [M[, 1], M[, 2], ..., M[, k]] ) | ( function )

Inputs two FpG-modules M and N with common group and characteristic. It returns the direct sum of M and N as an FpG-Module.

Alternatively, the function can input a list of FpG-modules with common group G. It returns the direct sum of the list.

Examples:

##### 21.1-3 FpGModule

`‣ FpGModule`( A, P ) | ( function )

`‣ FpGModule`( A, G, p ) | ( function )

Inputs a p-group P and a matrix A whose rows have length a multiple of the order of G. It returns the "canonical" FpG-module generated by the rows of A.

A small non-prime-power group G can also be input, provided the characteristic p is entered as a third input variable.

Examples: 1 , 2 , 3

##### 21.1-4 FpGModuleDualBasis

`‣ FpGModuleDualBasis`( M ) | ( function )

Inputs an FpG-module M. It returns a record R with two components:

- R.freeModule is the free module FG of rank one.

- R.basis is a list representing an F-basis for the module Hom_FG(M,FG). Each term in the list is a matrix A whose rows are vectors in FG such that M!.generators[i] ⟶ A[i] extends to a module homomorphism M ⟶ FG.

Examples:

##### 21.1-5 FpGModuleHomomorphism

`‣ FpGModuleHomomorphism`( M, N, A ) | ( function )

`‣ FpGModuleHomomorphismNC`( M, N, A ) | ( function )

Inputs FpG-modules M and N over a common p-group G. Also inputs a list A of vectors in the vector space spanned by N!.matrix. It tests that the function

M!.generators[i] ⟶ A[i]

extends to a homomorphism of FpG-modules and, if the test is passed, returns the corresponding FpG-module homomorphism. If the test is failed it returns fail.

The "NC" version of the function assumes that the input defines a homomorphism and simply returns the FpG-module homomorphism.

Examples:

##### 21.1-6 DesuspensionFpGModule

`‣ DesuspensionFpGModule`( M, n ) | ( function )

`‣ DesuspensionFpGModule`( R, n ) | ( function )

Inputs a positive integer n and and FpG-module M. It returns an FpG-module D^nM which is mathematically related to M via an exact sequence 0 ⟶ D^nM ⟶ R_n ⟶ ... ⟶ R_0 ⟶ M ⟶ 0 where R_∗ is a free resolution. (If G=Group(M) is of prime-power order then the resolution is minimal.)

Alternatively, the function can input a positive integer n and at least n terms of a free resolution R of M.

Examples:

##### 21.1-7 RadicalOfFpGModule

`‣ RadicalOfFpGModule`( M ) | ( function )

Inputs an FpG-module M with G a p-group, and returns the Radical of M as an FpG-module. (Ig G is not a p-group then a submodule of the radical is returned.

Examples:

##### 21.1-8 RadicalSeriesOfFpGModule

`‣ RadicalSeriesOfFpGModule`( M ) | ( function )

Inputs an FpG-module M and returns a list of FpG-modules that constitute the radical series for M.

Examples:

##### 21.1-9 GeneratorsOfFpGModule

`‣ GeneratorsOfFpGModule`( M ) | ( function )

Inputs an FpG-module M and returns a matrix whose rows correspond to a minimal generating set for M.

Examples:

##### 21.1-10 ImageOfFpGModuleHomomorphism

`‣ ImageOfFpGModuleHomomorphism`( f ) | ( function )

Inputs an FpG-module homomorphism f:M ⟶ N and returns its image f(M) as an FpG-module.

Examples:

##### 21.1-11 GroupAlgebraAsFpGModule

`‣ GroupAlgebraAsFpGModule`( G ) | ( function )

Inputs a p-group G and returns its mod p group algebra as an FpG-module.

Examples:

##### 21.1-12 IntersectionOfFpGModules

`‣ IntersectionOfFpGModules`( M, N ) | ( function )

Inputs two FpG-modules M, N arising as submodules in a common free module (FG)^n where G is a finite group and F the field of p-elements. It returns the FpG-module arising as the intersection of M and N.

Examples:

##### 21.1-13 IsFpGModuleHomomorphismData

`‣ IsFpGModuleHomomorphismData`( M, N, A ) | ( function )

Inputs FpG-modules M and N over a common p-group G. Also inputs a list A of vectors in the vector space spanned by N!.matrix. It returns true if the function

M!.generators[i] ⟶ A[i]

extends to a homomorphism of FpG-modules. Otherwise it returns false.

Examples:

##### 21.1-14 MaximalSubmoduleOfFpGModule

`‣ MaximalSubmoduleOfFpGModule`( M ) | ( function )

Inputs an FpG-module M and returns one maximal FpG-submodule of M.

Examples:

##### 21.1-15 MaximalSubmodulesOfFpGModule

`‣ MaximalSubmodulesOfFpGModule`( M ) | ( function )

Inputs an FpG-module M and returns the list of maximal FpG-submodules of M.

Examples:

##### 21.1-16 MultipleOfFpGModule

`‣ MultipleOfFpGModule`( w, M ) | ( function )

Inputs an FpG-module M and a list w:=[g_1 , ..., g_t] of elements in the group G=M!.group. The list w can be thought of as representing the element w=g_1 + ... + g_t in the group algebra FG, and the function returns a semi-echelon matrix B which is a basis for the vector subspace wM .

Examples:

##### 21.1-17 ProjectedFpGModule

`‣ ProjectedFpGModule`( M, k ) | ( function )

Inputs an FpG-module M of ambient dimension n|G|, and an integer k between 1 and n. The module M is a submodule of the free module (FG)^n . Let M_k denote the intersection of M with the last k summands of (FG)^n . The function returns the image of the projection of M_k onto the k-th summand of (FG)^n . This image is returned an FpG-module with ambient dimension |G|.

Examples:

##### 21.1-18 RandomHomomorphismOfFpGModules

`‣ RandomHomomorphismOfFpGModules`( M, N ) | ( function )

Inputs two FpG-modules M and N over a common group G. It returns a random matrix A whose rows are vectors in N such that the function

M!.generators[i] ⟶ A[i]

extends to a homomorphism M ⟶ N of FpG-modules. (There is a problem with this function at present.)

Examples:

##### 21.1-19 Rank

`‣ Rank`( f ) | ( function )

Inputs an FpG-module homomorphism f:M ⟶ N and returns the dimension of the image of f as a vector space over the field F of p elements.

Examples: 1

##### 21.1-20 SumOfFpGModules

`‣ SumOfFpGModules`( M, N ) | ( function )

Inputs two FpG-modules M, N arising as submodules in a common free module (FG)^n where G is a finite group and F the field of p-elements. It returns the FpG-Module arising as the sum of M and N.

Examples:

##### 21.1-21 SumOp

`‣ SumOp`( f, g ) | ( function )

Inputs two FpG-module homomorphisms f,g:M ⟶ N with common sorce and common target. It returns the sum f+g:M ⟶ N . (This operation is also available using "+".

Examples:

##### 21.1-22 VectorsToFpGModuleWords

`‣ VectorsToFpGModuleWords`( M, L ) | ( function )

Inputs an FpG-module M and a list L=[v_1,... ,v_k] of vectors in M. It returns a list L'= [x_1,...,x_k] . Each x_j=[[W_1,G_1],...,[W_t,G_t]] is a list of integer pairs corresponding to an expression of v_j as a word

v_j = g_1*w_1 + g_2*w_1 + ... + g_t*w_t

where

g_i=Elements(M!.group)[G_i]

w_i=GeneratorsOfFpGModule(M)[W_i] .

Examples:

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