The Automorphism Group of a Free Group#

AutomorphismGroup(F): GrpFP -> GrpAuto#: This intrinsic constructs the automorphism group of a free group \(F\).

InverseAutomorphismFreeGroup(F, Q): GrpFP, SeqEnum -> GrpAutoElt#: Given a free group \(F\) and a sequence \(Q\) of images of its generators, return the automorphism of \(F\) defined by \(Q\).

WhiteheadReduction(F, Q): GrpFP, SeqEnum -> BoolElt, SeqEnum, GrpAutoElt#: Perform Whitehead reduction of a sequence \(Q\) of elements of a free group \(F\). Return true if the list \(Q\) is part of a free basis, the Whitehead reduction, and corresponding automorphism of \(F\).

Example: Free Aut (ex-baab12)#

We find the automorphism group of the free group \(F\) of rank \(3\).

> F := FreeGroup(3);
> A, B := AutomorphismGroup(F);
> A;
A group of automorphisms of GrpFP: F
Generators:
    Automorphism of GrpFP: F which maps:
        F.1 |--> F.2
        F.2 |--> F.1
        F.3 |--> F.3
    Automorphism of GrpFP: F which maps:
        F.1 |--> F.2
        F.2 |--> F.3
        F.3 |--> F.1
    Automorphism of GrpFP: F which maps:
        F.1 |--> F.1^-1
        F.2 |--> F.2
        F.3 |--> F.3
    Automorphism of GrpFP: F which maps:
        F.1 |--> F.1 * F.2
        F.2 |--> F.2
        F.3 |--> F.3
> B;
Finitely presented group B on 4 generators
Relations
    B.1^2 = Id(B)
    B.2^3 = Id(B)
    (B.2 * B.1)^2 = Id(B)
    B.3^2 = Id(B)
    B.3^-1 * B.1^-1 * B.2^-1 * B.3 * B.2 * B.1 = Id(B)
    B.3^-1 * B.2^-1 * B.3^-1 * B.2 * B.3 * B.2^-1 * B.3 * B.2 = Id(B)
    B.3^-1 * B.2^-1 * B.1^-1 * B.2 * B.3 * B.2^-1 * B.1 * B.2 = Id(B)
    B.4^-1 * B.3^-1 * B.4^-1 * B.3^-1 * B.4 * B.3 * B.4 * B.3 = Id(B)
    B.4^-1 * B.1^-1 * B.2^-1 * B.3^-1 * B.4^-1 * B.3^-1 * B.2 * B.1^-1 * B.4 *
    B.1 * B.2^-1 * B.3 * B.4 * B.3 * B.2 * B.1 = Id(B)
    (B.1 * B.3 * B.1 * B.4)^2 = Id(B)
    B.4^-1 * B.1 * B.4 * B.1 * B.3 * B.4 * B.3 * B.1 * B.3 = Id(B)
    B.4^-1 * B.2^-1 * B.1^-1 * B.2 * B.1^-1 * B.2^-1 * B.4 * B.2 * B.1 * B.2^-1
    * B.1 * B.2 = Id(B)
    B.4^-1 * B.2^-2 * B.3^-1 * B.2^2 * B.4 * B.2^-2 * B.3 * B.2^2 = Id(B)
    B.4^-1 * B.1^-1 * B.2^-1 * B.1^-1 * B.2 * B.1^-1 * B.4^-1 * B.1^-1 * B.2^-1
    * B.1^-1 * B.2 * B.1^-1 * B.4 * B.1 * B.2^-1 * B.1 * B.2 * B.1 * B.4 * B.1 *
    B.2^-1 * B.1 * B.2 * B.1 = Id(B)
    B.1 * B.2^-1 * B.4 * B.2 * B.1 * B.2^-1 * B.4 * B.2 * B.4 * B.2^-1 * B.4^-1
    * B.2 * B.4^-1 = Id(B)

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The first return value is the automomorphism group given as a group of mappings. The second return value is the automorphism group as an fp-group. A different type of group is needed in order to return the presentation of the automomorphism group.

Example: Free Aut2 (ex-1a5108)#

In this example the automorphism group of a subgroup of the free group \(F\) of rank 3 is found.

> F<a, b, c> := FreeGroup(3);
> H<x, y> := sub< F | a, b>;
> K := H^(a*b*c);
> K;
Finitely presented group K on 2 generators
Generators as words in group F
    K.1 = c^-1 * b^-1 * a * b * c
    K.2 = c^-1 * b^-1 * a^-1 * b * a * b * c
> A, B := AutomorphismGroup(K);
> A;
A group of automorphisms of GrpFP: K
Generators:
    Automorphism of GrpFP: K which maps:
        K.1 |--> K.2
        K.2 |--> K.1
    Automorphism of GrpFP: K which maps:
        K.1 |--> K.1^-1
        K.2 |--> K.2
    Automorphism of GrpFP: K which maps:
        K.1 |--> K.1 * K.2
        K.2 |--> K.2
> B;
Finitely presented group B on 3 generators
Relations
    B.1^2 = Id(B)
    B.2^2 = Id(B)
    (B.2 * B.1)^4 = Id(B)
    B.3^-1 * B.2^-1 * B.3^-1 * B.2^-1 * B.3 * B.2 * B.3 * B.2 = Id(B)
    (B.1 * B.2 * B.1 * B.3)^2 = Id(B)
    (B.3 * B.1 * B.2)^3 = Id(B)

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