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 , 4

##### 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 \(M!.generators[i] \longrightarrow A[i]\) extends to a module homomorphism \(M \longrightarrow FG\).

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

\( M!.generators[i] \longrightarrow A[i]\)

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 \( 0 \longrightarrow D^nM \longrightarrow R_n \longrightarrow \ldots \longrightarrow R_0 \longrightarrow M \longrightarrow 0\) where \(R_\ast\) 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 \(f:M \longrightarrow N\) 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

\( M!.generators[i] \longrightarrow A[i]\)

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 \(w=g_1 + \ldots + g_t\) 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

\( M!.generators[i] \longrightarrow A[i]\)

extends to a homomorphism \(M \longrightarrow N\) 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 \(f:M \longrightarrow N\) and returns the dimension of the image of \(f\) as a vector space over the field \(F\) of \(p\) elements.

Examples: 1 , 2 , 3

##### 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 \(f,g:M \longrightarrow N\) with common sorce and common target. It returns the sum \(f+g:M \longrightarrow N\) . (This operation is also available using "+".

Examples:

##### 21.1-22 VectorsToFpGModuleWords

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

Inputs an \(FpG\)-module \(M\) and a list \(L=[v_1,\ldots ,v_k]\) 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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