Automorphisms

The elements of an automorphism group are automorphisms of the base group, so Magma treats them as both homomorphisms and group elements. Thus they may be applied to elements and subgroups of the base group as a homomorphism, or they may be multiplied and have inverses taken as group elements. Of course, these last two operations are also homomorphism operations, being composition and the usual inverse of a bijection. Elements of an automorphism group have Magma type GrpAutoElt.

A . i : GrpAuto, RngIntElt -> GrpAutoElt

The i-th generator for A. A negative subscript indicates that the inverse of the generator is to be created. A.0 is Identity(A).

Identity(A) : GrpAuto -> GrpAutoElt

Id(A) : GrpAuto -> GrpAutoElt

A ! 1 : GrpAuto, RngIntElt -> GrpAutoElt

The identity element of A.

A ! f : GrpAuto, Map -> GrpAutoElt

If f is any automorphism of the base group of A, represented as a Magma map, return the element of A corresponding to f. An error will result if f is not in the group generated by the generators of A. This uses ClassAction to test for membership.

Order(f) : GrpAutoElt -> RngIntElt

The order of f.

f * g : GrpAutoElt, GrpAutoElt -> GrpAutoElt

The product of the automorphisms f and g. As maps this is their composite: first apply f, then apply g.

f ^ n : GrpAutoElt, RngIntElt -> GrpAutoElt

The nth power of f.

(g_1, ..., g_r) : GrpAutoElt, ..., GrpAutoElt -> GrpAutoElt

Left-normed commutator of the automorphisms g_1, ..., g_r.

g eq h : GrpAutoElt, GrpAutoElt -> BoolElt

Given automorphisms g and h belonging to the same automorphism group, return true if g and h are the same element, false otherwise.

g ne h : GrpAutoElt, GrpAutoElt -> BoolElt

Given automorphisms g and h belonging to the same automorphism group, return false if g and h are the same element, true otherwise.

IsInner(f) : GrpAutoElt -> BoolElt, GrpElt

Returns true if f is an inner automorphism of the base group, false otherwise. If f is inner, then an element of the base group with conjugation action equal to the action of f is also returned.

Example GrpAuto_autogp-elts (H29E1)

> G := SmallGroup(81, 10);     
> G;
GrpPC : G of order 81 = 3^4
PC-Relations:
  G.1^3 = G.4, 
  G.2^3 = G.4^2, 
  G.2^G.1 = G.2 * G.3, 
  G.3^G.1 = G.3 * G.4
> A := AutomorphismGroup(G);
> #A;
486
> Ngens(A);
5
> IsInner(A.3);
false
> Order(A.3);
3
> A.3;
Automorphism of GrpPC : G of order 3 which maps:
  G.1 |--> G.1
  G.2 |--> G.2 * G.4^2
  G.3 |--> G.3
  G.4 |--> G.4
> A.3*A.4;
Automorphism of GrpPC : G which maps:
  G.1 |--> G.1
  G.2 |--> G.2 * G.3 * G.4^2
  G.3 |--> G.3 * G.4
  G.4 |--> G.4
> (A.3*A.4)^3;      
Automorphism of GrpPC : G which maps:
  G.1 |--> G.1
  G.2 |--> G.2
  G.3 |--> G.3
  G.4 |--> G.4
> $1 eq Id(A);
true