This intrinsic constructs the automorphism group of a free group F.
Given a free group F and a sequence Q of images of its generators,
return the automorphism of F defined by Q.
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.
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)
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.
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)
V2.28, 13 July 2023