Automorphisms

The following functions construct the standard automorphisms of a group of Lie type, as described in [Carter, 1972] (except for the graph automorphism of \(G_2\)). In many cases, including the finite groups, every automorphism is a product of these standard automorphisms.

Basic Functionality

AutomorphismGroup(G): GrpLie → GrpLieAuto

Automorphism group of a group of Lie type \(G\).

IdentityAutomorphism(G): GrpLie → GrpLieAutoElt

One(A): GrpLieAuto → GrpLieAutoElt

Id(A): GrpLieAuto → GrpLieAutoElt

The identity automorphism of the group of Lie type \(G\).

Mapping(a): GrpLieAutoElt → Map

The map object associated with the automorphism \(a\).

Automorphism(m): Map → GrpLieAutoElt

Given a map object \(m\) from \(G\) to \(G\), which is an isomorphism, returns the associated automorphism as an automorphism of a group of Lie type.

h * g: GrpLieAutoElt, GrpLieAutoElt → GrpLieAutoElt

The composition of the group of Lie type automorphisms \(h\) and \(g\).

h \^ n: GrpLieAutoElt, RngIntElt → GrpLieAutoElt

The \(n\)th power of the group of Lie type automorphism \(h\).

g \^ h: GrpLieAutoElt, GrpLieAutoElt → GrpLieAutoElt

The conjugate \(h^{-1}gh\), where \(g\) and \(h\) are group of Lie type automorphisms \(g\) and \(h\)

Domain(A): GrpLieAuto → GrpLie

Codomain(A): GrpLieAuto → GrpLie

Domain(h): GrpLieAutoElt → GrpLie

Codomain(h): GrpLieAutoElt → GrpLie

Domain or codomain of an automorphism of a group of Lie type or of the group of automorphisms.

Constructing Special Automorphisms

InnerAutomorphism(G, x): GrpLie, GrpLieElt → Map

The inner automorphism taking \(g\in G\) to \(g^x\), where \(x\) is an element of the group of Lie type \(G\).

DiagonalAutomorphism(G, v): GrpLie, ModTupRngElt → Map

The diagonal automorphism of the semisimple group of Lie type \(G\) given by the vector \(v\). Let \(n\) be the semisimple rank of \(G\) and let \(k\) be its base field. Then \(v\) must be a vector in \(k^n\) with every component nonzero. The function returns the automorphism given by the character \(\chi\) defined by \(\chi(\alpha_i)=v_i\), where \(\alpha_i\) is the \(i\)th simple root. Since our groups are algebraic, a diagonal automorphism is just a special case of an inner automorphism.

GraphAutomorphism(G, p): GrpLie, GrpPermElt → Map

DiagramAutomorphism(G, p): GrpLie, GrpPermElt → Map

SimpleSigns : Any                          Default: 1

The graph automorphism of the group of Lie type \(G\) given by the permutation \(p\). The permutation must act on the indices of simple roots of \(G\) or the indices of all roots of \(G\). The graph automorphism of the group of type \(G_2\) has not been implemented yet.

The optional parameter SimpleSigns can be used to specify the signs corresponding to each simple root. This should either be a sequence of integers \(\pm1\), or a single integer \(\pm1\).

FieldAutomorphism(G, sigma): GrpLie, Map → Map

The field automorphism of the group of Lie type \(G\) induced by \(\sigma\), an element of the automorphism group of the base field of \(G\)

RandomAutomorphism(G): GrpLie → GrpLieAutoElt

Random(A): GrpLieAuto → GrpLieAutoElt

A random element in \(A\), the automorphism group of the group of Lie type \(G\).

DualityAutomorphism(G): GrpLie → GrpLieAutoElt

The duality automorphism of \(G\). This is an automorphism that takes every unipotent term \(x_r(t)\) to \(x_s(\pm t)\), where \(s=\) Negative(RootDatum(G),r)).

FrobeniusMap(G, q): GrpLie, RngIntElt → GrpLieAutoElt

The Frobenius automorphism of the finite group of Lie type \(G\) gotten by \(q\)th powers in the base field. The integer \(q\) must be a power of the characteristic of the base field of \(G\).

Operations and Properties of Automorphisms

DecomposeAutomorphism(h): GrpLieAutoElt → GrpLieAutoElt, GrpLieAutoElt

GrpLieAutoElt, Rec

Given a group of Lie type automorphism \(h\), this returns a field automorphism \(f\), a graph automorphism \(g\) and an inner automorphism \(i\) such that \(h=fgi\). This only works for groups defined over finite fields. The algorithm is due to Scott Murray and Sergei Haller.

IsAlgebraic(h): GrpLieAutoElt → BoolElt

Returns true if and only if the automorphism \(h\) is algebraic.

Example: Automorphism

Some automorphisms of \(B_2(4)\) The automorphism of \(B_2(4)\) whose stabiliser is \({}^2\!B_2(4)\) is constructed by the following code.

> G := GroupOfLieType("B2", GF(4));
> A := AutomorphismGroup(G);
> A!1 eq IdentityAutomorphism(G);
true
%%a> assert $1;
> g := GraphAutomorphism(G, Sym(2)!(1,2));
> g;
Automorphism of Group of Lie type B2 over Finite field of size 2^2
given by: Mapping from: Group of Lie type  to Group of Lie type
given by a rule
Decomposition:
  Mapping from: GF(2^2) to GF(2^2) given by a rule,
  (1, 2),
  1
> sigma := iso< GF(4) -> GF(4) | x :-> x^2, x :-> x^2 >;
> h := FieldAutomorphism(G, sigma) * g;
> h in A;
true
%%a> assert $1;
> f,g,i := DecomposeAutomorphism(h);
> assert f*g*i eq h;