Page 79 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 79
mož Moravec: Some Topics in the Theory of Finite Groups 67
The group of symmetries of a regular n -gon is called a dihedral group D2n . If a de-
notes the rotation around the center by the angle 2π/n, and b the reflection over a cho-
sen diagonal, then the elements of D2n can be written uniquely in the form a k b where
0 ≤ k < n and ∈ {0, 1}. Thus |D2n | = 2n . The group D2n has a presentation
D2n = 〈a ,b | a n = 1, b 2 = 1, a b = a −1〉.
In GAP, one can construct dihedral groups directly by
gap> G := DihedralGroup( 6 );
gap> Order( G );
6
Another way is to present it by generators and relations. This is done by first con-
structing a free group on two generators and then factor out the relations.
gap> F := FreeGroup( "a", "b" );
gap> AssignGeneratorVariables(F);
#I Assigned the global variables [ a, b ]
gap> H := F / [ a^3, b^2, a^b / a^(-1) ];
gap> StructureDescription( H );
"S3"
The last command tells us that D6 =∼ S3. We can compare both constructions of D6
above and see that they are not identical objects in GAP, yet they are isomorphic:
gap> H = G;
false
gap> IsomorphismGroups(G, H);
[ f1, f2 ] -> [ b, a ]
The reason is that GAP represents D6 in two different ways, first as a pc group and
then as an fp group. The reader should consult GAP’s manual for further details.
3.2.3 Automorphisms
An automorphism of a group G is an isomorphism G to itself. There are special types
of automorphisms called conjugations or inner automorphisms; they are of the form
c g : x → g −1x g .
Proposition 3.2.11 Let G be a group.
(a) The set Aut(G ) of all automorphisms of G is a group under composition (from the
right). This is the automorphism group of G .
(b) The set Inn(G ) of all inner automorphisms of G is a normal subgroup of AutG . This
is called the inner automorphism group of G .
The group of symmetries of a regular n -gon is called a dihedral group D2n . If a de-
notes the rotation around the center by the angle 2π/n, and b the reflection over a cho-
sen diagonal, then the elements of D2n can be written uniquely in the form a k b where
0 ≤ k < n and ∈ {0, 1}. Thus |D2n | = 2n . The group D2n has a presentation
D2n = 〈a ,b | a n = 1, b 2 = 1, a b = a −1〉.
In GAP, one can construct dihedral groups directly by
gap> G := DihedralGroup( 6 );
gap> Order( G );
6
Another way is to present it by generators and relations. This is done by first con-
structing a free group on two generators and then factor out the relations.
gap> F := FreeGroup( "a", "b" );
gap> AssignGeneratorVariables(F);
#I Assigned the global variables [ a, b ]
gap> H := F / [ a^3, b^2, a^b / a^(-1) ];
gap> StructureDescription( H );
"S3"
The last command tells us that D6 =∼ S3. We can compare both constructions of D6
above and see that they are not identical objects in GAP, yet they are isomorphic:
gap> H = G;
false
gap> IsomorphismGroups(G, H);
[ f1, f2 ] -> [ b, a ]
The reason is that GAP represents D6 in two different ways, first as a pc group and
then as an fp group. The reader should consult GAP’s manual for further details.
3.2.3 Automorphisms
An automorphism of a group G is an isomorphism G to itself. There are special types
of automorphisms called conjugations or inner automorphisms; they are of the form
c g : x → g −1x g .
Proposition 3.2.11 Let G be a group.
(a) The set Aut(G ) of all automorphisms of G is a group under composition (from the
right). This is the automorphism group of G .
(b) The set Inn(G ) of all inner automorphisms of G is a normal subgroup of AutG . This
is called the inner automorphism group of G .
