Poder automorfismos que no es interior

Question

Quiero saber un ejemplo de un pedido pequeño, no abelian $p$grupo $G$ con una potencia automorphism que no es interior, es decir, un automorphism de la forma $g\mapsto g^k$ todos los $g\in G$, pero no interior.

En los ejemplos que inicialmente estaba considerando, que los mapas se $g\mapsto g^{-1}$ que nunca será automorfismos de no abelian grupos.

Cualquier buen ejemplo de esto? Gracias por su interés.

Answer 1

Un simple cálculo de la BRECHA de rendimientos de respuestas. He utilizado este programa:

isNonInnerPowerAuto := function(G, k)
    local gens, imgs, hom;
    gens := GeneratorsOfGroup(G);
    imgs := List(gens, x->x^k);
    hom := GroupHomomorphismByImages(G, G, gens, imgs);
    if hom = fail then return false; fi;
    if not IsBijective(hom) then return false; fi;
    if IsInnerAutomorphism(hom) then return false; fi;
    return ForAll(G, g -> g^hom = g ^ k);
end;

for n in [6..127] do
    if not IsPrimePowerInt(n) then continue; fi;
    for i in [1..NrSmallGroups(n)] do
        G := SmallGroup(n,i);
        if IsAbelian(G) then continue; fi;
        for k in [2..Exponent(G)-2] do
            if isNonInnerPowerAuto(G,k) then
                 Print("for group ", IdGroup(G),", ",k, " is an auto, non-inner\n");
            fi;
        od;
    od;
od;

Lo que resultó en esta salida:

for group [ 32, 5 ], 5 is an auto, non-inner
for group [ 32, 12 ], 5 is an auto, non-inner
for group [ 32, 17 ], 5 is an auto, non-inner
for group [ 32, 17 ], 13 is an auto, non-inner
for group [ 32, 38 ], 5 is an auto, non-inner
for group [ 64, 3 ], 5 is an auto, non-inner
for group [ 64, 4 ], 5 is an auto, non-inner
for group [ 64, 5 ], 5 is an auto, non-inner
for group [ 64, 17 ], 5 is an auto, non-inner
for group [ 64, 27 ], 5 is an auto, non-inner
for group [ 64, 27 ], 13 is an auto, non-inner
for group [ 64, 29 ], 5 is an auto, non-inner
for group [ 64, 29 ], 9 is an auto, non-inner
for group [ 64, 29 ], 13 is an auto, non-inner
for group [ 64, 30 ], 5 is an auto, non-inner
for group [ 64, 30 ], 13 is an auto, non-inner
for group [ 64, 31 ], 9 is an auto, non-inner
for group [ 64, 44 ], 5 is an auto, non-inner
for group [ 64, 44 ], 9 is an auto, non-inner
for group [ 64, 44 ], 13 is an auto, non-inner
for group [ 64, 51 ], 5 is an auto, non-inner
for group [ 64, 51 ], 9 is an auto, non-inner
for group [ 64, 51 ], 13 is an auto, non-inner
for group [ 64, 51 ], 21 is an auto, non-inner
for group [ 64, 51 ], 25 is an auto, non-inner
for group [ 64, 51 ], 29 is an auto, non-inner
for group [ 64, 86 ], 5 is an auto, non-inner
for group [ 64, 87 ], 5 is an auto, non-inner
for group [ 64, 89 ], 5 is an auto, non-inner
for group [ 64, 103 ], 5 is an auto, non-inner
for group [ 64, 105 ], 5 is an auto, non-inner
for group [ 64, 112 ], 5 is an auto, non-inner
for group [ 64, 114 ], 5 is an auto, non-inner
for group [ 64, 115 ], 5 is an auto, non-inner
for group [ 64, 116 ], 5 is an auto, non-inner
for group [ 64, 117 ], 5 is an auto, non-inner
for group [ 64, 126 ], 5 is an auto, non-inner
for group [ 64, 127 ], 5 is an auto, non-inner
for group [ 64, 184 ], 5 is an auto, non-inner
for group [ 64, 184 ], 13 is an auto, non-inner
for group [ 64, 185 ], 5 is an auto, non-inner
for group [ 64, 185 ], 9 is an auto, non-inner
for group [ 64, 185 ], 13 is an auto, non-inner
for group [ 64, 248 ], 5 is an auto, non-inner
for group [ 81, 3 ], 4 is an auto, non-inner
for group [ 81, 3 ], 7 is an auto, non-inner
for group [ 81, 4 ], 4 is an auto, non-inner
for group [ 81, 4 ], 7 is an auto, non-inner
for group [ 81, 6 ], 4 is an auto, non-inner
for group [ 81, 6 ], 7 is an auto, non-inner
for group [ 81, 6 ], 13 is an auto, non-inner
for group [ 81, 6 ], 16 is an auto, non-inner
for group [ 81, 6 ], 22 is an auto, non-inner
for group [ 81, 6 ], 25 is an auto, non-inner
for group [ 81, 14 ], 4 is an auto, non-inner
for group [ 81, 14 ], 7 is an auto, non-inner

El más pequeño de los ejemplos son de orden 32. Tenga en cuenta que $k$ es siempre coprime para el orden del grupo, por lo que si se induce un automorphism, ciertamente, no es interior.

Para trabajar con cualquiera de ellos con la mano, usted puede pedir BRECHA para una presentación:

gap> G := SmallGroup(32,5);
<pc group of size 32 with 5 generators>
gap> StructureDescription(G);
"(C8 x C2) : C2"
gap> gfp:=Image(IsomorphismFpGroup(G));
<fp group of size 32 on the generators [ F1, F2, F3, F4, F5 ]>
gap> RelatorsOfFpGroup(gfp);
[ F1^2*F4^-1, F2^-1*F1^-1*F2*F1*F3^-1, F3^-1*F1^-1*F3*F1, F4^-1*F1^-1*F4*F1,
  F5^-1*F1^-1*F5*F1, F2^2, F3^-1*F2^-1*F3*F2, F4^-1*F2^-1*F4*F2,
  F5^-1*F2^-1*F5*F2, F3^2, F4^-1*F3^-1*F4*F3, F5^-1*F3^-1*F5*F3, F4^2*F5^-1,
  F5^-1*F4^-1*F5*F4, F5^2 ]

O, pida un poder-la conjugación de la presentación (esto omite trivial de la conjugación de las relaciones, por ejemplo, g4^g3 = g4; mirar la documentación de PrintPcpPresentation para los detalles; ah sí, y tenga en cuenta que es parte de la policíclicos paquete, que es, sin embargo, se instala y cargado por defecto de forma regular en una BRECHA de la instalación):

gap> PrintPcpPresentation(PcGroupToPcpGroup(G));
g1^2 = g4
g2^2 = id
g3^2 = id
g4^2 = g5
g5^2 = id
g2 ^ g1 = g2 * g3

O pedir una isomorfo permutación grupo (llevo dos, uno es el regular la representación, una un poco más pequeña:

gap> H := Image(IsomorphismPermGroup(G));;
gap> SmallGeneratingSet(H);
[ (1,18,26,17,6,29,14,28)(2,13,30,23,10,3,20,31)(4,27,16,7,15,32,5,19)(8,24,22,
    12,21,11,9,25), (1,24)(2,28)(3,15)(4,13)(5,31)(6,11)(7,21)(8,19)(9,32)(10,
    17)(12,26)(14,25)(16,23)(18,30)(20,29)(22,27) ]
gap> SmallGeneratingSet(Image(SmallerDegreePermutationRepresentation(H)));
[ (1,2,4,7,5,8,11,14)(3,6,9,12,10,13,15,16), (2,6)(7,12)(8,13)(14,16),
  (1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,15)(14,16) ]