Page 80 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

3.2 Basic notions and examples

(c) Inn(G ) ∼= G /Z (G ).

The proof is straightforward and we leave it as an exercise. The group Out(G ) =

Aut(G )/ Inn(G ) is the outer automorphism group of G . Note that its elements are not

automorphisms, but rather right cosets Inn(G )α, where α ∈ Aut(G ).

GAP can deal with automorphisms very naturally:

gap> G := DihedralGroup( 12 );

gap> A := AutomorphismGroup( G );

gap> Elements( A );
[ [ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f2, f1*f3^2, f1 ],

[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f2*f3^2, f1*f3, f1 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1, f1*f2*f3, f1*f2 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f3, f1*f2*f3^2, f1*f2 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f2, f1*f3^2, f1*f3 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f2*f3, f1, f1*f3 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f3, f1*f2*f3^2, f1*f2*f3 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f3^2, f1*f2, f1*f2*f3 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f2*f3, f1, f1*f3^2 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f2*f3^2, f1*f3, f1*f3^2 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1, f1*f2*f3, f1*f2*f3^2 ],
[ f1*f3, f1*f2*f3^2, f1*f2*f3 ] -> [ f1*f3^2, f1*f2, f1*f2*f3^2 ] ]
gap> StructureDescription( A );
"D12"
gap> inn := InnerAutomorphismsAutomorphismGroup( A );

gap> Order( inn );
6
gap> IsomorphismGroups( inn, G / Center( G ) );
CompositionMapping( [ (2,6)(3,5), (1,3,5)(2,4,6), (1,5,3)(2,6,4) ] ->
[ f1, f2^2, f2 ], )

Next we compute some automorphism groups:

Proposition 3.2.12 AutCn ∼= Cφ(n), where φ is Euler’s totient function.

PROOF. Let Cn = 〈g 〉 and take α ∈ AutG . Then g α = g i for some 0 ≤ i ≤ n − 1, and since

〈g i 〉 = Cn , this can only happen if gcd(i , n ) = 1. Conversely take an endomorphism α of

Cn with g α = g i , where gcd(i , n ) = 1. Then it is elementary to see that α is an automor-

phism. Thus the map AutCn → × given by α → i is an isomorphism of groups. This
n

proves the result.

Proposition 3.2.13 Aut(C n ) =∼ GL(n , p ).
p

PROOF. This follows from the fact that C n is an n -dimensional vector space over GF(p ).
p