# Creation of a Matrix Group ## Construction of the General Linear Group ### `GeneralLinearGroup(n, R): RngIntElt, Rng -> GrpMat` ### `GL(n, R): RngIntElt, Rng -> GrpMat` Given an integer $n \geq 1$ and a ring $R$, create the generic matrix group, i.e. the general linear group ${\operatorname{GL}}(n, R)$. Initially, only a structure table is created for ${\operatorname{GL}}(n, R)$, so that, in particular, generators are not defined. This function is normally used to provide a context for the creation of elements and subgroups of ${\operatorname{GL}}(n, R)$. If structural computation is attempted with the group created by `GeneralLinearGroup(n, R)`, then generators will be created where possible. At present, this is only permitted in the cases in which $R$ is a finite field. ### `Example: Create (ex-59ac81)` We define the general linear group ${\operatorname{GL}}(3, K)$, where $K$ is the finite field ${\bf F}_{4}$. ```magma > K := FiniteField(4); > GL34 := GeneralLinearGroup(3, K); > GL34; GL(3, GF(2, 2)) ``` ## Construction of a Matrix Group Element Throughout this subsection we shall assume that the matrix group $G$ is defined over the ring $R$. ### `elt< G | L >: GrpMat, List(RngElt) -> GrpMatElt` Given a matrix group $G$ defined as a subgroup of ${\operatorname{GL}}(n, R)$, and the list $L$ of expressions $a_{ij}$ ($1 \leq i, j \leq n$), defining elements of the ring $R$, construct the $n\times n$ matrix $$ \begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix} $$ Unless $G$ is known to be the generic matrix group of degree $n$, the matrix will be tested for membership of $G$, and if $g$ is not an element of $G$, the function will fail. If $g$ does lie in $G$, $g$ will have $G$ as its parent. Since the membership test may involve constructing a base and strong generating set for $G$, this constructor may occasionally be very costly. Hence a matrix $g$ should be defined as an element of a subgroup of the generic group only when membership of $G$ is required by subsequent operations involving $g$. ### `G ! Q: GrpMat, [ RngElt ] -> GrpMatElt` Given the sequence $Q$ of expressions $a_{ij}$ ($1 \leq i, j \leq n$), defining elements of the ring $R$, construct the $n\times n$ matrix $$ \begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix} $$ This matrix will have $G$ as its parent structure. As in the case of the `elt`-constructor, the operation will fail if $g$ is not an element of $G$, and the same observations concerning the cost of membership testing apply. ### `ElementToSequence(g): GrpMatElt -> [ RngElt ]` ### `Eltseq(g): GrpMatElt -> [ RngElt ]` Given an $n \times n$ matrix $g = (a_{ij}), 1 \le i, j \le n$, where $a_{ij}$ is an element of the ring $R$, construct the sequence $$ [ a_{11}, \ldots, a_{1n}, a_{21}, \ldots, a_{2n}, \ldots a_{n1}, \ldots, a_{nn}] $$ of $n^2$ elements of the ring $R$. ### `Identity(G): GrpMat -> GrpMatElt` ### `Id(G): GrpMat -> GrpMatElt` ### `G ! 1: GrpMat, RngIntElt -> GrpMatElt` Construct the identity matrix in the matrix group $G$. ### `Example: Matrices (ex-da01c4)` The different constructions are illustrated by the following code, which assigns to each of the variables $x$ and $y$ an element of ${\operatorname{GL}}(3, 4)$. ```magma > K := FiniteField(4); > GL34 := GeneralLinearGroup(3, K); > x := elt; > x; [w 0 1] [0 1 0] [1 0 1] > y := GL34 ! [w,0,1, 0,1,0, 1,0,1]; > y; [w 0 1] [0 1 0] [1 0 1] > GL34 ! 1; [1 0 0] [0 1 0] [0 0 1] ``` ## Construction of a General Matrix Group ### `MatrixGroup< n, R | L >: RngIntElt, Rng, List -> GrpMat` Construct the matrix group $G$ of degree $n$ over the ring $R$ generated by the matrices defined by the list $L$. A term of the list $L$ must be an object of one of the following types: **(a)** A sequence of $n^2$ elements of $R$ defining a matrix of ${\operatorname{GL}}(n, R)$; **(b)** A set or sequence of sequences of type (a); **(c)** An element of ${\operatorname{GL}}(n, R)$; **(d)** A set or sequence of elements of ${\operatorname{GL}}(n, R)$; **(e)** A subgroup of ${\operatorname{GL}}(n, R)$; **(f)** A set or sequence of subgroups of ${\operatorname{GL}}(n, R)$. Each element or group specified by the list must belong to the *same* generic matrix group. The group $G$ will be constructed as a subgroup of some group which contains each of the elements and groups specified in the list. The generators of $G$ consist of the elements specified by the terms of the list $L$ together with the stored generators for groups specified by terms of the list. Repetitions of an element and occurrences of the identity element are removed. The `MatrixGroup` constructor is shorthand for the two statements: ``` GL := GeneralLinearGroup(n, R); G := sub< GL | L >; ``` where `sub< ... >` is the subgroup constructor described in the next subsection. ### `Example: Constructor (ex-383e20)` We use the `MatrixGroup` constructor to define a small subgroup of ${\operatorname{GL}}(3, 4)$. ```magma > K := FiniteField(4); > H := MatrixGroup< 3, K | [1,w,0, 0,1,0, 1,w^2,1], [w,0,0, 0,1,0, 0,0,w] >; > H; MatrixGroup(3, GF(2, 2)) Generators: [ 1 w 0] [ 0 1 0] [ 1 w^2 1] [w 0 0] [0 1 0] [0 0 w] > Order(H); 96 ``` ### `Example: GL Sylow (ex-0a504f)` We present a function which will construct the Sylow $p$-subgroup of ${\operatorname{GL}}(n, K)$, where $K$ is a finite field of characteristic $p$. ```magma > GLSyl := function(n, K) > R := MatrixRing(K, n); > e := func< i, j | MatrixUnit(R, i, j) >; > return MatrixGroup< n, K | { R!1 + a*e(i,j) : a in K, j in [i+1], > i in [1 .. n - 1] | a ne 0 } >; > end function; > T := GLSyl(3, GF(8)); > FactoredOrder(T); [ <2, 9> ] > FactoredOrder(GL(3, GF(8))); [ <2, 9>, <3, 2>, <7, 3>, <73, 1> ] ``` ## Changing Rings ### `ChangeRing(G, S): GrpMat, Rng -> GrpMat, Map` Given a matrix group $G$ with base ring $R$, construct a new matrix group $H$ with base ring $S$ derived from $G$ by coercing entries of the generators of $G$ from $R$ into $S$. ### `ChangeRing(G, S, f): GrpMat, Rng, Map -> GrpMat, Map` Given a matrix group $G$ with base ring $R$, construct a new matrix group $H$ with base ring $S$ derived from $G$ by applying $f$ to the entries of the generators of $G$. ### `RestrictField(G, S): GrpMat, FldFin -> GrpMat, Map` Given a matrix group $G$ with base ring $K$, a finite field, and $S$ a subfield of $K$, construct the matrix group $H$ with base ring $S$ obtained by restricting the scalars of the components of elements of $G$ into $S$, together with the restriction map from $G$ onto $H$. ### `ExtendField(G, L): GrpMat, FldFin -> GrpMat, Map` Given a matrix group $G$ with base ring $K$, a finite field, and $L$ an extension of $K$, construct the matrix group $H$ with base ring $L$ obtained by lifting the components of elements of $G$ into $L$, together with the inclusion homomorphism from $G$ into $H$. ## Coercion between Matrix Structures A square non-singular matrix may be defined as an element of any of the following structures: - A subring of the complete matrix ring $M_{n}(R)$; - A subgroup of the general linear group ${\operatorname{GL}}(n, R)$; - A submodule of the matrix module $M^{(m \times n)}(R)$. The coercion operator may be used to transfer matrices between any two of these three structures. ### `R ! g: AlgMat, GrpMatElt -> RngMatElt` Transfer the matrix $g$ from a group into a matrix ring $R$. ### `G ! r: GrpMat, AlgMatElt -> GrpMatElt` Transfer the matrix $r$ from a ring into a matrix group $G$. ### `M ! g: ModMatRng, GrpMatElt -> ModMatRngElt` Transfer the matrix $g$ from a group into a matrix module $M$. ### `G ! m: GrpMat, ModMatRngElt -> GrpMatElt` Transfer the matrix $m$ from a module into a matrix group $G$. ### `ProjectionLocalization(g, pi): GrpMatElt, Map -> GrpMatElt` For a square matrix $g$ with elements inside a localization $R_{(p)}$ of a ring $R$ (so all denominators are prime to $p$), given a projection map $\pi : R \to R/p$, returns the image of $g$ under the map induced by $\pi$. ## Accessing Associated Structures The functions in this group provide access to basic information stored for a matrix group $G$. ### `G . i: GrpMat, RngIntElt -> GrpMatElt` The $i$-th defining generator for the matrix group $G$. A negative subscript indicates that the inverse of the generator is to be created. The $0$th generator `G.0` is `Identity(G)`. ### `Degree(G): GrpMat -> RngIntElt` The degree of the matrix group $G$. ### `Generators(G): GrpMat -> { GrpMatElt }` A set containing the defining generators for the matrix group $G$. ### `NumberOfGenerators(G): GrpMat -> RngIntElt` ### `Ngens(G): GrpMat -> RngIntElt` The number of defining generators for the matrix group $G$. ### `CoefficientRing(G): GrpMat -> Rng` ### `BaseRing(G): GrpMat -> Rng` The coefficient ring for the matrix group $G$. ### `RSpace(G): GrpMat -> ModTupRng` Given a matrix group $G$ of degree $n$ defined over a ring $R$, return the space $R^{(n)}$, where the action is multiplication by elements of $R$, i.e. scalar action. ### `VectorSpace(G): GrpMat -> ModTupFld` Given a matrix group $G$ of degree $n$ defined over a field $K$, return the space $K^{(n)}$, where the action is multiplication by elements of $K$, i.e. scalar action. ### `GModule(G): GrpMat -> ModGrp` The natural $R[G]$-module for the matrix group $G$. ### `Generic(G): GrpMat -> GrpMat` The generic group containing the matrix group $G$, i.e. the general linear group in which $G$ is naturally embedded. ### `Parent(G): GrpMatElt -> GrpMat` The power structure for the group $G$ (the set consisting of all matrix groups).